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, que establece que por un punto dado se puede trazar una única recta paralela a una recta dada. Points A, B, and D are not collinear. In spherical geometry, there are no parallel lines. The sum of the angles in a triangle is always greater than 180°. Glossary/Glosario S139 S139 ������������������������������������������������������������������������� ENGLISH noncoplanar (p. 6) Points that do not lie on the same plane. SPANISH EXAMPLES no coplanar Puntos que no se encuentran en el mismo plano. numerator (p. 451) The top number of a fraction, which tells how many parts of a whole are being considered. numerador El número superior de una fracción, que indica la cantidad de partes de un entero que se consideran. T, U, V, and S are not coplanar. The numerator of 3 _ 7 is 3. O oblique cone (p. 690) A cone whose axis is not perpendicular to the base. cono oblicuo Cono cuyo eje no es perpendicular a la base. oblique cylinder (p. 681) A cylinder whose axis is not perpendicular to the bases. cilindro oblicuo Cilindro cuyo eje no es perpendicular a las bases. oblique prism (p. 680) A prism that has at least one nonrectangular lateral face. prisma oblicuo Prisma que tiene por lo menos una cara lateral no rectangular. obtuse angle (p. 21) An angle that measures greater than 90° and less than 180°. ángulo obtuso Ángulo que mide más de 90° y menos de 180°. obtuse triangle (p. 216) A triangle with one obtuse angle. triángulo obtusángulo Triángulo con un ángulo obtuso. octagon (p. 382) An eight-sided polygon. octágono Polígono de ocho lados. octahedron (p. 669) A polyhedron with eight faces. octaedro Poliedro con |
ocho caras. one-point perspective (p. 662) A perspective drawing with one vanishing point. perspectiva de un punto Dibujo en perspectiva con un punto de fuga. opposite rays (p. 7) Two rays that have a common endpoint and form a line. rayos opuestos Dos rayos que tienen un extremo común y forman una recta. EF and EG are opposite rays. S140 S140 Glossary/Glosario ����������������������� ENGLISH opposite reciprocal (p. 184) The opposite of the reciprocal of a number. The opposite reciprocal of a is - 1 __ a. order of rotational symmetry (p. 857) The number of times a figure with rotational symmetry coincides with itself as it rotates 360°. SPANISH EXAMPLES recíproco opuesto Opuesto del recíproco de un número. El recíproco opuesto de a es - 1 __ a. The opposite reciprocal of 2 _ 3 is -3 _ 2 orden de simetría de rotación Cantidad de veces que una figura con simetría de rotación coincide consigo misma cuando rota 360°. Order of rotational symmetry: 4 ordered pair (p. 42) A pair of numbers (x, y) that can be used to locate a point on a coordinate plane. The first number x indicates the distance to the left or right of the origin, and the second number y indicates the distance above or below the origin. par ordenado Par de números (x, y) que se pueden utilizar para ubicar un punto en un plano cartesiano. El primer número indica la distancia a la izquierda o derecha del origen y el segundo número indica la distancia hacia arriba o hacia abajo del origen. ordered triple (p. 671) A set of three numbers that can be used to locate a point (x, y, z) in a threedimensional coordinate system. tripleta ordenada Conjunto de tres números que se pueden utilizar para ubicar |
un punto (x, y, z) en un sistema de coordenadas tridimensional. origin (p. 42) The intersection of the x- and y-axes in a coordinate plane. The coordinates of the origin are (0, 0). origen Intersección de los ejes x e y en un plano cartesiano. Las coordenadas de origen son (0, 0). orthocenter of a triangle (p. 316) The point of concurrency of the three altitudes of a triangle. ortocentro de un triángulo Punto de intersección de las tres alturas de un triángulo. P is the orthocenter. orthographic drawing (p. 661) A drawing that shows a threedimensional object in which the line of sight for each view is perpendicular to the plane of the picture. dibujo ortográfico Dibujo que muestra un objeto tridimensional en el que la línea visual para cada vista es perpendicular al plano de la imagen. Glossary/Glosario S141 S141 ����������������������������������������������������������������������������������������� ENGLISH outcome (p. 628) A possible result of a probability experiment. SPANISH resultado Resultado posible de un experimento de probabilidades. EXAMPLES In the experiment of rolling a number cube, the possible outcomes are 1, 2, 3, 4, 5, and 6. P paragraph proof (p. 120) A style of proof in which the statements and reasons are presented in paragraph form. demostración con párrafos Tipo de demostración en la cual los enunciados y las razones se presentan en forma de párrafo. parallel lines (p. 146) Lines in the same plane that do not intersect. líneas paralelas Líneas rectas en el mismo plano que no se cruzan. parallel planes (p. 146) Planes that do not intersect. planos paralelos Planos que no se cruzan. r ǁ s parallel vectors (p. 561) Vectors with the same or opposite direction. vectores paralelos Vectores con direcci |
ón igual u opuesta. Plane AEF and plane CGH are parallel planes. parallelogram (p. 391) A quadrilateral with two pairs of parallel sides. parallelogram method (p. 561) A method of adding two vectors by drawing a parallelogram using the vectors as two of the consecutive sides; the sum is a vector along the diagonal of the parallelogram. paralelogramo Cuadrilátero con dos pares de lados paralelos. método del paralelogramo Método mediante el cual se suman dos vectores dibujando un paralelogramo, utilizando los vectores como dos de los lados consecutivos; el resultado de la suma es un vector a lo largo de la diagonal del paralelogramo. parent function (p. 221) The simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent function. función madre La función más básica que tiene las características distintivas de una familia. Las funciones de la misma familia son transformaciones de su función madre. Pascal’s triangle (p. 883) A triangular arrangement of numbers in which every row starts and ends with 1 and each other number is the sum of the two numbers above it. triángulo de Pascal Arreglo triangular de números en el cual cada fila comienza y termina con 1 y cada uno de los otros números es la suma de los dos números que están encima de él. S142 S142 Glossary/Glosario f (x) = x 2 is the parent function for g (x) = x 2 + 4 and h (x) = 5 (x + 2) 2 - 3. ��������������������������������������������������������������������������������������������������������������� ENGLISH pentagon (p. 382) A five-sided polygon. SPANISH pentágono Polígono de cinco lados. EXAMPLES perimeter (p. 36) The sum of the side lengths of a closed plane figure. perímetro Suma |
de las longitudes de los lados de una figura plana cerrada. perpendicular (p. 146) Intersecting to form 90° angles, denoted by ⊥. perpendicular Que se cruza para formar ángulos de 90°, expresado por ⊥. Perimeter = 18 + 6 + 18 + 6 = 48 ft m ⊥ n perpendicular bisector of a segment (p. 172) A line perpendicular to a segment at the segment’s midpoint. mediatriz de un segmento Línea perpendicular a un segmento en el punto medio del segmento. perpendicular lines (p. 146) Lines that intersect at 90° angles. líneas perpendiculares Líneas que se cruzan en ángulos de 90°. ℓ is the perpendicular bisector of ̶̶ AB. m ⊥ n perspective drawing (p. 662) A drawing in which nonvertical parallel lines meet at a point called a vanishing point. Perspective drawings can have one or two vanishing points. dibujo en perspectiva Dibujo en el cual las líneas paralelas no verticales se encuentran en un punto denominado punto de fuga. Los dibujos en perspectiva pueden tener uno o dos puntos de fuga. pi (p. 37) The ratio of the circumference of a circle to its diameter, denoted by the Greek letter π (pi). The value of π is irrational, often approximated by 3.14 or 22 __ 7. pi Razón entre la circunferencia de un círculo y su diámetro, expresado por la letra griega π (pi). El valor de π es irracional y por lo general se aproxima a 3.14 ó 22 __ 7. If a circle has a diameter of 5 inches and a circumference of C inches, then C __ = π, or C = 5π inches, or 5 about 15.7 inches. plane (p. 6) An undefined term in geometry, it is a flat surface that has no thickness and extends forever. plano Término indefinido en geometría; un plano es una superficie plana que no tiene grosor y |
se extiende infinitamente. plane symmetry (p. 858) A threedimensional figure that can be divided into two congruent reflected halves by a plane has plane symmetry. simetría de plano Una figura tridimensional que se puede dividir en dos mitades congruentes reflejadas por un plano tiene simetría de plano. Platonic solid (p. 669) One of the five regular polyhedra: a tetrahedron, a cube, an octahedron, a dodecahedron, or an icosahedron. sólido platónico Uno de los cinco poliedros regulares: tetraedro, cubo, octaedro, dodecaedro o icosaedro. plane R or plane ABC Glossary/Glosario S143 S143 ������������������������������������������������ ENGLISH point (p. 6) An undefined term in geometry, it names a location and has no size. SPANISH EXAMPLES punto Término indefinido de la geometría que denomina una ubicación y no tiene tamaño. point P point matrix (p. 846) A matrix that represents the coordinates of the vertices of a polygon. The first row of the matrix consists of the xcoordinates of the points, and the second row consists of the y-coordinates. matriz de puntos Matriz que representa las coordenadas de los vértices de un polígono. La primera fila de la matriz contiene las coordenadas x de los puntos y la segunda fila contiene las coordenadas y. ⎡ ⎢ ⎣ 1 2 -2 0 3 -4 ⎤ ⎥ ⎦ point of concurrency (p. 307) A point where three or more lines coincide. punto de concurrencia Punto donde se cruzan tres o más líneas. point of tangency (p. 746) The point of intersection of a circle or sphere with a tangent line or plane. punto de tangencia Punto de intersección de un círculo o esfera con una línea o plano tangente. point |
-slope form (p. 190) y - y 1 = m (x - x 1 ), where m is the slope and ( x 1, y 1 ) is a point on the line. polar axis (p. 808) In a polar coordinate system, the horizontal ray with the pole as its endpoint that lies along the positive x-axis. forma de punto y pendiente (y - y 1 ) = m (x - x 1 ), donde m es la pendiente y ( x 1, y 1 ) es un punto en la línea. eje polar En un sistema de coordenadas polares, el rayo horizontal, cuyo extremo es el polo, que se encuentra a lo largo del eje x positivo. polar coordinate system (p. 808) A system in which a point in a plane is located by its distance r from a point called the pole, and by the measure of a central angle θ. sistema de coordenadas polares Sistema en el cual un punto en un plano se ubica por su distancia r de un punto denominado polo y por la medida de un ángulo central θ. S144 S144 Glossary/Glosario ��������������������������������������������������������������������������������������������������������������������� ENGLISH pole (p. 808) The point from which distances are measured in a polar coordinate system. SPANISH polo Punto desde el que se miden las distancias en un sistema de coordenadas polares. EXAMPLES polygon (p. 98) A closed plane figure formed by three or more segments such that each segment intersects exactly two other segments only at their endpoints and no two segments with a common endpoint are collinear. polígono Figura plana cerrada formada por tres o más segmentos tal que cada segmento se cruza únicamente con otros dos segmentos sólo en sus extremos y ningún segmento con un extremo común a otro es colineal con éste. polyhedron (p. 670) A closed three-dimensional figure formed by four or more polygons that intersect only at their edges. pol |
iedro Figura tridimensional cerrada formada por cuatro o más polígonos que se cruzan sólo en sus aristas. postulate (p. 7) A statement that is accepted as true without proof. Also called an axiom. postulado Enunciado que se acepta como verdadero sin demostración. También denominado axioma. preimage (p. 50) The original figure in a transformation. imagen original Figura original en una transformación. primes (p. 50) Symbols used to label the image in a transformation. apóstrofos Símbolos utilizados para identificar la imagen en una transformación. A′B′C′ prism (p. 713) A polyhedron formed by two parallel congruent polygonal bases connected by lateral faces that are parallelograms. probability (p. 237) A number from 0 to 1 (or 0% to 100%) that is the measure of how likely an event is to occur. prisma Poliedro formado por dos bases poligonales congruentes paralelas conectadas por caras laterales que son paralelogramos. probabilidad Número entre 0 y 1 (o entre 0% y 100%) que describe cuán probable es que ocurra un suceso. A bag contains 3 red marbles and 4 blue marbles. The probability of randomly choosing a red marble is 3 __. 7 proof (p. 104) An argument that uses logic to show that a conclusion is true. demostración Argumento que se vale de la lógica para probar que una conclusión es verdadera. proof by contradiction (p. 322) See indirect proof. demostración por contradicción Ver demostración indirecta. Glossary/Glosario S145 S145 ������������������������������������ ENGLISH proportion (p. 455) A statement = c __ that two ratios are equal; a __. d b SPANISH proporción Ecuación que establece = c __ que dos razones son iguales; a __. d b EXAMPLES = 4 _ 2 _ 6 3 pyramid (p. 654) A polyhedron |
formed by a polygonal base and triangular lateral faces that meet at a common vertex. pirámide Poliedro formado por una base poligonal y caras laterales triangulares que se encuentran en un vértice común. Pythagorean triple (p. 349) A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2. Tripleta de Pitágoras Conjunto de tres números cabales distintos de cero a, b y c tal que 3, 4, 5 quadrant (p. 42) One of the four regions into which the x- and y-axes divide the coordinate plane. cuadrante Una de las cuatro regiones en las que los ejes x e y dividen el plano cartesiano. quadrilateral (p. 98) A four-sided polygon. cuadrilátero Polígono de cuatro lados. R radial symmetry (p. 857) See rotational symmetry. simetría radial Ver simetría de rotación. radical symbol (p. 346) The symbol √ used to denote a root. The symbol is used alone to indicate a square root or with an index, n √ , to indicate the nth root. símbolo de radical Símbolo √ que se utiliza para expresar una raíz. Puede utilizarse solo para indicar una raíz cuadrada, o con un índice, n √ , para indicar la enésima raíz. √ 36 = 6 3 √ 27 = 3 radicand (p. 346) The expression under a radical sign. radicando Número o expresión debajo del signo de radical. Expression: √ x + 3 Radicand: x + 3 radius of a circle (p. 37) A segment whose endpoints are the center of a circle and a point on the circle; the distance from the center of a circle to any point on the circle. radio de un círculo Segmento cuyos extremos son el cent |
ro y un punto del círculo; distancia desde el centro de un círculo hasta cualquier punto de éste. radius of a cone (p. 681) The distance from the center of the base of the cone to any point on the base. radio de un cono Distancia desde el centro de la base del cono hasta un punto cualquiera de la base. S146 S146 Glossary/Glosario ���������������������������������������������������������������� EXAMPLES ENGLISH radius of a cylinder (p. 690) The distance from the center of the base of the cylinder to any point on the base. SPANISH radio de un cilindro Distancia desde el centro de la base del cilindro hasta un punto cualquiera de la base. radius of a sphere (p. 714) A segment whose endpoints are the center of a sphere and any point on the sphere; the distance from the center of a sphere to any point on the sphere. radio de una esfera Segmento cuyos extremos son el centro de una esfera y cualquier punto sobre la esfera; distancia desde el centro de una esfera hasta cualquier punto sobre la esfera. rate of change (p. 97) A ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. tasa de cambio Razón que compara la cantidad de cambio de la variable dependiente con la cantidad de cambio de la variable independiente. Rate of change = change in y __ change in x = 6 _ 4 = 3 _ 2 ratio (p. 454) A comparison of two quantities by division. razón Comparación de dos cantidades mediante una división. rational number (p. 80) A number that can be written in the form a __, b where a and b are integers and b ≠ 0. ray (p. 7) A part of a line that starts at an endpoint and extends forever in one direction. número racional Número que se puede expresar como a __, d |
onde a y b b son números enteros y b ≠ 0. ̶ 3, - 2 _ 3, 1.75, 0. 3, 0 rayo Parte de una recta que comienza en un extremo y se extiende infinitamente en una dirección. rectangle (p. 408) A quadrilateral with four right angles. rectángulo Cuadrilátero con cuatro ángulos rectos. reduction (p. 873) A dilation with a scale factor greater than 0 but less than 1. In a reduction, the image is smaller than the preimage. reducción Dilatación con un factor de escala mayor que 0 pero menor que 1. En una reducción, la imagen es más pequeña que la imagen original. reference angle (p. 570) For an angle in standard position, the reference angle is the positive acute angle formed by the terminal side of the angle and the x-axis. ángulo de referencia Dado un ángulo en posición estándar, el ángulo de referencia es el ángulo agudo positivo formado por el lado terminal del ángulo y el eje x. Glossary/Glosario S147 S147 ������������������������������������������������������������������ ENGLISH reflection (p. 50) A transformation across a line, called the line of reflection, such that the line of reflection is the perpendicular bisector of each segment joining each point and its image. SPANISH EXAMPLES reflexión Transformación sobre una línea, denominada la línea de reflexión. La línea de reflexión es la mediatriz de cada segmento que une un punto con su imagen. reflection symmetry (p. 856) See line symmetry. simetría de reflexión Ver simetría axial. regular polygon (p. 382) A polygon that is both equilateral and equiangular. polígono regular Polígono equilátero de ángulos iguales. regular polyhedron (p. 669) A polyhedron in which all faces are congruent regular polygons and the same number |
of faces meet at each vertex. See also Platonic solid. poliedro regular Poliedro cuyas caras son todas polígonos regulares congruentes y en el que el mismo número de caras se encuentran en cada vértice. Ver también sólido platónico. regular pyramid (p. 689) A pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. pirámide regular Pirámide cuya base es un polígono regular y cuyas caras laterales son triángulos isósceles congruentes. regular tessellation (p. 864) A repeating pattern of congruent regular polygons that completely covers a plane with no gaps or overlaps. teselado regular Patrón que se repite formado por polígonos regulares congruentes que cubren completamente un plano sin dejar espacios y sin superponerse. relation (p. 389) A set of ordered pairs. relación Conjunto de pares ordenados. (0, 5), (0, 4), (2, 3), (4, 0) remote interior angle (p. 225) An interior angle of a polygon that is not adjacent to the exterior angle. ángulo interno remoto Ángulo interno de un polígono que no es adyacente al ángulo externo. The remote interior angles of ∠4 are ∠1 and ∠2 resultant vector (p. 561) The vector that represents the sum of two given vectors. vector resultante Vector que representa la suma de dos vectores dados. rhombus (p. 409) A quadrilateral with four congruent sides. rombo Cuadrilátero con cuatro lados congruentes. right angle (p. 21) An angle that measures 90°. ángulo recto Ángulo que mide 90°. S148 S148 Glossary/Glosario ��������������������������������������� ENGLISH right cone (p. 690) A cone whose axis is perpendicular to the base. SPANISH |
EXAMPLES cono regular Cono cuyo eje es perpendicular a la base. right cylinder (p. 681) A cylinder whose axis is perpendicular to its bases. cilindro regular Cilindro cuyo eje es perpendicular a sus bases. right prism (p. 680) A prism whose lateral faces are all rectangles. prisma regular Prisma cuyas caras laterales son todas rectángulos. right triangle (p. 216) A triangle with one right angle. triángulo rectángulo Triángulo con un ángulo recto. rigid transformation (p. 824) See isometry. rise (p. 182) The difference in the y-values of two points on a line. transformación rígida Ver isometría. distancia vertical Diferencia entre los valores de y de dos puntos de una línea. For the points (3, -1) and (6, 5), the rise is 5 - (-1) = 6. rotation (p. 50) A transformation about a point P, also known as the center of rotation, such that each point and its image are the same distance from P. All of the angles with vertex P formed by a point and its image are congruent. rotación Transformación sobre un punto P, también conocido como el centro de rotación, tal que cada punto y su imagen estén a la misma distancia de P. Todos los ángulos con vértice P formados por un punto y su imagen son congruentes. rotational symmetry (p. 857) A figure that can be rotated about a point by an angle less than 360° so that the image that coincides with the preimage has rotational symmetry. simetría de rotación Una figura que puede rotarse alrededor de un punto en un ángulo menor de 360° de forma tal que la imagen coincide con la imagen original que tenga simetría de rotación. run (p. 182) The difference in the x-values of two points on a line. distancia horizontal Diferencia entre los valores de x de dos puntos |
de una línea. S same-side interior angles (p. 147) For two lines intersected by a transversal, a pair of angles that lie on the same side of the transversal and between the two lines. ángulos internos del mismo lado Dadas dos rectas cortadas por una transversal, el par de ángulos ubicados en el mismo lado de la transversal y entre las dos rectas. Order of rotational symmetry: 4 For the points (3, -1) and (6, 5), the run is 6 - 3 = 3. ∠2 and ∠3 are same-side interior angles. Glossary/Glosario S149 S149 ��������������������������������������������������� ENGLISH SPANISH sample space (p. 628) The set of all possible outcomes of a probability experiment. espacio muestral Conjunto de todos los resultados posibles de un experimento de probabilidades. EXAMPLES in the experiment of rolling a number cube, the sample space is {1, 2, 3, 4, 5, 6}. scalar multiplication of a vector (p. 566) The process of multiplying a vector by a constant. multiplicación escalar de un vector Proceso por el cual se multiplica un vector por una constante. 3〈-8, 1〉 = 〈-24, 3〉 scale (p. 489) The ratio between two corresponding measurements. escala Razón entre dos medidas correspondientes. 1 cm : 5 mi A blueprint is an example of a scale drawing. Scale factor: 2 scale drawing (p. 489) A drawing that uses a scale to represent an object as smaller or larger than the actual object. dibujo a escala Dibujo que utiliza una escala para representar un objeto como más pequeño o más grande que el objeto original. scale factor (p. 495) The multiplier used on each dimension to change one figure into a similar figure. factor de escala El multiplicador utilizado en cada dimensión para transformar una figura en una figura semejante. scale model (p. 456) A threedimensional model that uses a scale |
to represent an object as smaller or larger than the actual object. modelo a escala Modelo tridimensional que utiliza una escala para representar un objeto como más pequeño o más grande que el objeto real. scalene triangle (p. 217) A triangle with no congruent sides. triángulo escaleno Triángulo sin lados congruentes. scatter plot (p. 198) A graph with points plotted to show a possible relationship between two sets of data. diagrama de dispersión Gráfica con puntos dispersos para demostrar una relación posible entre dos conjuntos de datos. secant of a circle (p. 746) A line that intersects a circle at two points. secante de un círculo Línea que corta un círculo en dos puntos. S150 S150 Glossary/Glosario ��������������������������������������������������������������������������� ENGLISH SPANISH EXAMPLES secant of an angle (p. 532) In a right triangle, the ratio of the length of the hypotenuse to the length of the side adjacent to angle A. It is the reciprocal of the cosine function. secante de un ángulo En un triángulo rectángulo, la razón entre la longitud de la hipotenusa y la longitud del cateto adyacente al ángulo A. Es la inversa de la función coseno. secant segment (p. 793) A segment of a secant with at least one endpoint on the circle. segmento secante Segmento de una secante que tiene al menos un extremo sobre el círculo. sector of a circle (p. 764) A region inside a circle bounded by two radii of the circle and their intercepted arc. sector de un círculo Región dentro de un círculo delimitado por dos radios del círculo y por su arco abarcado. segment bisector (p. 16) A line, ray, or segment that divides a segment into two congruent segments. bisectriz de un segmento Línea, rayo o segment |
o que divide un segmento en dos segmentos congruentes. segment of a circle (p. 765) A region inside a circle bounded by a chord and an arc. segmento de un círculo Región dentro de un círculo delimitada por una cuerda y un arco. segment of a line (p. 7) A part of a line consisting of two endpoints and all points between them. segmento de una línea Parte de una línea que consiste en dos extremos y todos los puntos entre éstos. self-similar (p. 882) A figure that can be divided into parts, each of which is similar to the entire figure. autosemejante Figura que se puede dividir en partes, cada una de las cuales es semejante a la figura entera. semicircle (p. 756) An arc of a circle whose endpoints lie on a diameter. semicírculo Arco de un círculo cuyos extremos se encuentran sobre un diámetro. semiregular tessellation (p. 864) A repeating pattern formed by two or more regular polygons in which the same number of each polygon occur in the same order at every vertex and completely cover a plane with no gaps or overlaps. teselado semirregular Patrón que se repite formado por dos o más polígonos regulares en los que el mismo número de cada polígono se presenta en el mismo orden en cada vértice y que cubren un plano completamente sin dejar espacios vacíos ni superponerse. sec A = hypotenuse __ adjacent = 1 _ cos A ̶̶̶ NM is an external secant segment. ̶̶ JK is an internal secant segment. Glossary/Glosario S151 S151 ������������������������������������������� ENGLISH side of a polygon (p. 382) One of the segments that form a polygon. SPANISH EXAMPLES lado de un polígono Uno de los segmentos que forman un polígono. side of an angle (p |
. 20) One of the two rays that form an angle. lado de un ángulo Uno de los dos rayos que forman un ángulo. AB are AC and sides of ∠CAB. Sierpinski triangle (p. 882) A fractal formed from a triangle by removing triangles with vertices at the midpoints of the sides of each remaining triangle. triángulo de Sierpinski Fractal formado a partir de un triángulo al cual se le recortan triángulos cuyos vértices se encuentran en los puntos medios de los lados de cada triángulo restante. similar (p. 462) Two figures are similar if they have the same shape but not necessarily the same size. similar polygons (p. 462) Two polygons whose corresponding angles are congruent and whose corresponding sides are proportional. similarity ratio (p. 463) The ratio of two corresponding linear measurements in a pair of similar figures. similarity statement (p. 463) A statement that indicates that two polygons are similar by listing the vertices in the order of correspondence. semejantes Dos figuras con la misma forma pero no necesariamente del mismo tamaño. polígonos semejantes Dos polígonos cuyos ángulos correspondientes son congruentes y cuyos lados correspondientes son proporcionales. razón de semejanza Razón de dos medidas lineales correspondientes en un par de figuras semejantes. Similarity ratio: 3.5 _ 2.1 = 5 _ 3 enunciado de semejanza Enunciado que indica que dos polígonos son similares enumerando los vértices en orden de correspondencia. quadrilateral ABCD ∼ quadrilateral EFGH sine (p. 525) In a right triangle, the ratio of the length of the leg opposite ∠A to the length of the hypotenuse. seno En un triángulo rectángulo, razón entre la longitud del cateto opuesto a ∠A y la longitud de la hipoten |
usa. skew lines (p. 146) Lines that are not coplanar. líneas oblicuas Líneas que no son coplanares. S152 S152 Glossary/Glosario sin A = opposite __ hypotenuse AE and CD are skew lines. ��������������������������������������������������������������������������������� ENGLISH slant height of a regular pyramid (p. 689) The distance from the vertex of a regular pyramid to the midpoint of an edge of the base. SPANISH EXAMPLES altura inclinada de una pirámide regular Distancia desde el vértice de una pirámide regular hasta el punto medio de una arista de la base. slant height of a right cone (p. 690) The distance from the vertex of a right cone to a point on the edge of the base. altura inclinada de un cono regular Distancia desde el vértice de un cono regular hasta un punto en el borde de la base. slide (p. 50) See translation. deslizamiento Ver traslación. slope (p. 182) A measure of the steepness of a line. If ( x 1, y 1 ) and ( x 2, y 2 ) are any two points on the line, the slope of the line, known as m, is represented by the y 2 - y 1 _____ equation m = x 2 - x 1. pendiente Medida de la inclinación de una línea. Dados dos puntos ( ) en una línea, la pendiente de la línea, denominada m, se representa por la ecuación m = y 2 - y 1 _____ x 2 - x 1. slope-intercept form (p. 190) The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. forma de pendiente-intersección La forma de pendiente-intersección de una ecuación lineal es y = mx + b, donde m es la pendient |
e y b es la intersección con el eje y. solid (p. 654) A three-dimensional figure. cuerpo geométrico Figura tridimensional. solving a triangle (p. 535) Using given measures to find unknown angle measures or side lengths of a triangle. resolución de un triángulo Utilizar medidas dadas para descubrir las medidas desconocidas de los ángulos o las longitudes laterales de un triángulo. space (p. 671) The set of all points in three dimensions. espacio Conjunto de todos los puntos en tres dimensiones. special parallelogram (p. 410) A rectangle, rhombus, or square. paralelogramo especial Un rectángulo, rombo o cuadrado. special quadrilateral (p. 391) A parallelogram, rectangle, rhombus, square, kite, or trapezoid. cuadrilátero especial Un paralelogramo, rectángulo, rombo, cuadrado, cometa o trapecio. special right triangle (p. 356) A 45°-45°-90° triangle or a 30°-60°-90° triangle. triángulo rectángulo especial Triángulo de 45°-45°-90° o triángulo de 30°-60°-90°. y = -2x + 4 The slope is -2. The y-intercept is 4. Glossary/Glosario S153 S153 ��������������������������������������������������������������� ENGLISH sphere (p. 714) The set of points in space that are a fixed distance from a given point called the center of the sphere. SPANISH EXAMPLES esfera Conjunto de puntos en el espacio que se encuentran a una distancia fija de un punto determinado denominado centro de la esfera. spherical geometry (p. 726) A system of geometry defined on a sphere. A line is defined as a great circle of the sphere, and there are no parallel lines. geometría esférica Sistema de geometría definido sobre una esfer |
a. Una línea se define como un gran círculo de la esfera y no existen líneas paralelas. square (p. 410) A quadrilateral with four congruent sides and four right angles. cuadrado Cuadrilátero con cuatro lados congruentes y cuatro ángulos rectos. standard position (p. 687) An angle in standard position has its vertex at the origin and its initial side on the positive x-axis. posición estándar Ángulo cuyo vértice se encuentra en el origen y cuyo lado inicial se encuentra sobre el eje x positivo. straight angle (p. 21) A 180° angle. ángulo llano Ángulo que mide 180°. subtend (p. 772) A segment or arc subtends an angle if the endpoints of the segment or arc lie on the sides of the angle. subtender Un segmento o arco subtiende un ángulo si los extremos del segmento o arco se encuentran sobre los lados del ángulo. If D and F are the endpoints of an arc or chord, and E is a point ̶̶ ̶̶ DF, then ⁀ DF or DF is said not on to subtend ∠DEF. supplementary angles (p. 29) Two angles whose measures have a sum of 180°. ángulos suplementarios Dos ángulos cuyas medidas suman 180°. ∠3 and ∠4 are supplementary angles. surface area (p. 680) The total area of all faces and curved surfaces of a three-dimensional figure. área total Área total de todas las caras y superficies curvas de una figura tridimensional. Surface area = 2 (8) (12) + 2 (8) (6) + 2 (12) (6) = 432 cm 2 symmetry (p. 856) In the transformation of a figure such that the image coincides with the preimage, the image and preimage have symmetry. simetría En la transformación de una figura tal que la imagen coincide con la imagen original, la imagen y la imag |
en original tienen simetría. S154 S154 Glossary/Glosario ������������������������� ENGLISH symmetry about an axis (p. 858) In the transformation of a figure such that there is a line about which a three-dimensional figure can be rotated by an angle greater than 0° and less than 360° so that the image coincides with the preimage, the image and preimage have symmetry about an axis. SPANISH simetría axial En la transformación de una figura tal que existe una línea sobre la cual se puede rotar una figura tridimensional a un ángulo mayor que 0° y menor que 360° de forma que la imagen coincida con la imagen original, la imagen y la imagen original tienen simetría axial. EXAMPLES system of equations (p. 421) A set of two or more equations that have two or more variables. sistema de ecuaciones Conjunto de dos o más ecuaciones que contienen dos o más variables. 2x + 3y = -1 3x - 3y = 4 T tangent circles (p. 747) Two coplanar circles that intersect at exactly one point. If one circle is contained inside the other, they are internally tangent. If not, they are externally tangent. círculos tangentes Dos círculos coplanares que se cruzan únicamente en un punto. Si un círculo contiene a otro, son tangentes internamente. De lo contrario, son tangentes externamente. tangent of an angle (p. 525) In a right triangle, the ratio of the length of the leg opposite ∠A to the length of the leg adjacent to ∠A. tangente de un ángulo En un triángulo rectángulo, razón entre la longitud del cateto opuesto a ∠A y la longitud del cateto adyacente a ∠A. tangent segment (p. 794) A segment of a tangent with one endpoint on the circle. segmento tangente Segmento de una tangente con un extremo en el círculo. tangent |
of a circle (p. 746) A line that is in the same plane as a circle and intersects the circle at exactly one point. tangente de un círculo Línea que se encuentra en el mismo plano que un círculo y lo cruza únicamente en un punto. tangent of a sphere (p. 805) A line that intersects the sphere at exactly one point. tangente de una esfera Línea que toca la esfera únicamente en un punto. tan A = opposite _ adjacent ̶̶ BC is a tangent segment. Glossary/Glosario S155 S155 ������������������������� ENGLISH terminal point of a vector (p. 559) The endpoint of a vector. SPANISH EXAMPLES punto terminal de un vector Extremo de un vector. terminal side (p. 570) For an angle in standard position, the ray that is rotated relative to the positive x-axis. lado terminal Para un ángulo en posición estándar, el rayo que se rota en relación con el eje x positivo. tessellation (p. 863) A repeating pattern of plane figures that completely covers a plane with no gaps or overlaps. teselado Patrón que se repite formado por figuras planas que cubren completamente un plano sin dejar espacios libres y sin superponerse. tetrahedron (p. 669) A polyhedron with four faces. A regular tetrahedron has equilateral triangles as faces, with three faces meeting at each vertex. tetraedro Poliedro con cuatro caras. Las caras de un tetraedro regular son triángulos equiláteros y cada vértice es compartido por tres caras. theorem (p. 110) A statement that has been proven. teorema Enunciado que ha sido demostrado. theoretical probability (p. 214) The ratio of the number of equally likely outcomes in an event to the total number of possible outcomes. probabilidad teórica Razón entre el número de resultados igualmente probables de un suceso y el nú |
mero total de resultados posibles. In the experiment of rolling a number cube, the theoretical probability of rolling an odd number is 3 __ 6 = 1 __. 2 three-dimensional coordinate system (p. 671) A space that is divided into eight regions by an x-axis, a y-axis, and a z-axis. The locations, or coordinates, of points are given by ordered triples. sistema de coordenadas tridimensional Espacio dividido en ocho regiones por un eje x, un eje y un eje z. Las ubicaciones, o coordenadas, de los puntos son dadas por tripletas ordenadas. tick marks (p. 13) Marks used on a figure to indicate congruent segments. marcas “|” Marcas utilizadas en una figura para indicar segmentos congruentes. tiling (p. 862) See tessellation. teselación Ver teselado transformation (p. 50) A change in the position, size, or shape of a figure or graph. transformación Cambio en la posición, tamaño o forma de una figura o gráfica. S156 S156 Glossary/Glosario ����������������������������������������������������������������������������������������������������� ENGLISH SPANISH EXAMPLES translation (p. 50) A transformation that shifts or slides every point of a figure or graph the same distance in the same direction. translation symmetry (p. 863) A figure has translation symmetry if it can be translated along a vector so that the image coincides with the preimage. traslación Transformación en la que todos los puntos de una figura o gráfica se mueven la misma distancia en la misma dirección. simetría de traslación Una figura tiene simetría de traslación si se puede trasladar a lo largo de un vector de forma tal que la imagen coincida con la imagen original. transversal (p. 147) A line that intersects two coplanar lines at two different points. transversal Línea que corta dos líneas coplanares en |
dos puntos diferentes. trapezoid (p. 429) A quadrilateral with exactly one pair of parallel sides. triangle (p. 98) A three-sided polygon. trapecio Cuadrilátero con sólo un par de lados paralelos. triángulo Polígono de tres lados. triangle rigidity (p. 242) A property of triangles that states that if the side lengths of a triangle are fixed, the triangle can have only one shape. rigidez del triángulo Propiedad de los triángulos que establece que, si las longitudes de los lados de un triángulo son fijas, el triángulo puede tener sólo una forma. triangulation (p. 223) The method for finding the distance between two points by using them as vertices of a triangle in which one side has a known, or measurable, length. triangulación Método para calcular la distancia entre dos puntos utilizándolos como vértices de un triángulo en el cual un lado tiene una longitud conocida o medible. trigonometric ratio (p. 525) A ratio of two sides of a right triangle. razón trigonométrica Razón entre dos lados de un triángulo rectángulo. trigonometry (p. 514) The study of the measurement of triangles and of trigonometric functions and their applications. trigonometría Estudio de la medición de los triángulos y de las funciones trigonométricas y sus aplicaciones. trisect (p. 25) To divide into three equal parts. trisecar Dividir en tres partes iguales. truth table (p. 128) A table that lists all possible combinations of truth values for a statement and its components. tabla de verdad Tabla en la que se enumeran todas las combinaciones posibles de valores de verdad para un enunciado y sus componentes. sin A = a _ c ; cos A = b _ c ; tan A = a _ b ̶̶̶ AD is trisected. Glossary/Glosario S157 S157 |
��������������������������������������������� ENGLISH SPANISH EXAMPLES truth value (p. 82) A statement can have a truth value of true (T) or false (F). valor de verdad Un enunciado puede tener un valor de verdad verdadero (V) o falso (F). turn (p. 50) See rotation. giro Ver rotación. two-column proof (p. 111) A style of proof in which the statements are written in the left-hand column and the reasons are written in the right-hand column. demostración a dos columnas Estilo de demostración en la que los enunciados se escriben en la columna de la izquierda y las razones en la columna de la derecha. two-point perspective (p. 662) A perspective drawing with two vanishing points. perspectiva de dos puntos Dibujo en perspectiva con dos puntos de fuga. U undefined term (p. 6) A basic figure that is not defined in terms of other figures. The undefined terms in geometry are point, line, and plane. término indefinido Figura básica que no está definida en función de otras figuras. Los términos indefinidos en geometría son el punto, la línea y el plano. unit circle (p. 570) A circle with a radius of 1, centered at the origin. círculo unitario Círculo con un radio de 1, centrado en el origen. V vanishing point (p. 662) In a perspective drawing, a point on the horizon where parallel lines appear to meet. punto de fuga En un dibujo en perspectiva, punto en el horizonte donde todas las líneas paralelas parecen encontrarse. vector (p. 559) A quantity that has both magnitude and direction. vector Cantidad que tiene magnitud y dirección. Venn diagram (p. 80) A diagram used to show relationships between sets. diagrama de Venn Diagrama utilizado para mostrar la relación entre conjuntos. S158 S158 Glossary/Glosario �������� |
�������������������������������������������������������������������������������������� ENGLISH vertex angle of an isosceles triangle (p. 273) The angle formed by the legs of an isosceles triangle. SPANISH ángulo del vértice de un triángulo isósceles Ángulo formado por los catetos de un triángulo isósceles. EXAMPLES vertex of a cone (p. 654) The point opposite the base of the cone. vértice de un cono Punto opuesto a la base del cono. vertex of a graph (p. 95) A point on a graph. vértice de una gráfica Punto en una gráfica. vertex of a polygon (p. 382) The intersection of two sides of the polygon. vértice de un polígono La intersección de dos lados del polígono. A, B, C, D, and E are vertices of the polygon. vertex of a pyramid (p. 689) The point opposite the base of the pyramid. vértice de una pirámide Punto opuesto a la base de la pirámide. vertex of a three-dimensional figure (p. 654) The point that is the intersection of three or more faces of the figure. vertex of a triangle (p. 216) The intersection of two sides of the triangle. vértice de una figura tridimensional Punto que representa la intersección de tres o más caras de la figura. vértice de un triángulo Intersección de dos lados del triángulo. vertex of an angle (p. 20) The common endpoint of the sides of the angle. vértice de un ángulo Extremo común de los lados del ángulo. vertical angles (p. 30) The nonadjacent angles formed by two intersecting lines. ángulos opuestos por el vértice Ángulos no adyacentes formados por dos rectas que se cruzan. A, B, and C are vertices of △ABC. A is the vertex |
of ∠CAB. ∠1 and ∠3 are vertical angles. ∠2 and ∠4 are vertical angles. volume (p. 697) The number of nonoverlapping unit cubes of a given size that will exactly fill the interior of a three-dimensional figure. volumen Cantidad de cubos unitarios no superpuestos de un determinado tamaño que llenan exactamente el interior de una figura tridimensional. Volume = (3) (4) (12) = 144 ft 3 Glossary/Glosario S159 S159 ���������������������������������������������������������������������������������������������� ENGLISH SPANISH EXAMPLES W whole number (p. 80) The set of natural numbers and zero. número cabal Conjunto de los números naturales y cero. 0, 1, 2, 3, 4, 5, … X x-axis (p. 42) The horizontal axis in a coordinate plane. eje x Eje horizontal en un plano cartesiano. Y y-axis (p. 42) The vertical axis in a coordinate plane. eje y Eje vertical en un plano cartesiano. Z z-axis (p. 671) The third axis in a three-dimensional coordinate system. eje z Tercer eje en un sistema de coordenadas tridimensional. S160 S160 Glossary/Glosario ����������������� Index A AA (angle-angle) similarity, 470 AAS (angle-angle-side) congruence, 254 proof of, 254 Absolute error, S73 Absolute value, 21, S61 equations, see Equations expressions, see Expressions Accuracy, S72 Acute angle, 21 Acute triangle, 216 Addition of vectors, 561–562 Addition Property of Equality, 104 Addition Property of Inequality, 330 Adjacent angles, 28 Adjacent arcs, 757 Adjustable parallels, 836 Advertising, 499 Agriculture, 765 Ahmes Papyrus, 41 Algebra, 40, 100, 109, 115, 124, 156, 159, 162, 164, 217, 224–225, 244, 249, 261, 274, 301–303, 308, 312, 316, 318, 325–326, 341, 355, 384–385, 387 |
, 393, 396, 399, 403, 409, 415, 424, 430, 433–434, 523, 531–532, 590, 597, 605, 620–621, 627, 634–635, 675, 677, 704, 749, 753, 758–759, 762, 775, 777, 788, 792–794, 798–799, 803, 805, 860, S56–S69 The review and development of algebra skills is found throughout this book. absolute value, 19, 21, S61 equations, see Equations expressions, see Expressions binomials, multiplying, 40 coordinate plane, 42, 43, 361, 393, 397, 400, 402, 405, 410, 420–423, 434, 435, S56 area in the, 616–619 circles in the, 799–801 dilations in the, 495–497, 874 distance in the, 43–46 graphing in the, 42 lines in the, 190–193 midpoint in the, 43–46 parallelograms in the, 393 perimeter in the, 616–619 reflections in the, 826 rotations in the, 840 similarity in the, 495–497 strategies for positioning figures in the, 267 transformations in the, 50–52 translations in the, 832 determining whether lines are parallel, perpendicular, or neither, 184 direct variation, 161, 501, S62 equation(s) of circles, 799–805 finding, 805 of a horizontal line, 190 of a line, 303, 304, 305, 306, 308, 311, 312, 313, 315–318, 339 literal, 588, 590, S59 quadratic, 266 solving, 25 linear, 11, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 29, 31, 32, 33, 34, 38, 39, 40, 41, 44, 104–109, 124, 155, 156, 158, 159, 219, 220, 221, 227, 228, 230, 235, 236, 237, 245, 246, 249, 259, 264, 265, 272, 276, 277, 301, 302, 304, 305, 318, 320, 325, 337, 349, 352, 353, 354, 355, 384, 385, 386, 387, 388, |
392, 393, 395, 396, 397, 403, 405, 406, 409, 410, 412, 423, 425, 430, 432, 433, 434, 751, 753, 760, 761, 762, 776, 777, 778, 779, 786, 787, 788, 789, 795, 796, 797, 798, 807, 814, S58 literal, 41, 169, S59 quadratic, 27, 228, 230, 235, 237, 246, 259, 277, 326, 349, 350, 352, 353, 354, 355, 388, 415, 430, 432, 433, 434, 494, 752, 771, 777, 796, 797, 798, 804, 814, S66 radical, 49, S68 systems of, 125, 158, 159, 176, 177, 194, 805 by elimination, 152–153, 157, 193, S67 by graphing, 8, 193, 195, 196 by substitution, 316–318, 396, S67 of a vertical line, 190 writing, 11, 18, 25 linear, 19, 31, 32, 33, 34, 38, 39, 40, 41 of lines, 190–197 literal, 41 expressions, 40 evaluating, 19, 162, 164–167, 334, 336, 384, 392, 393, 395, 399, 402, 405, 408–410, 412, 429, 432, 433 simplifying, 19, 36 writing, 229, 788, S57 factoring, to solve quadratic equations, S66 finding slope, see Slope functions, 11, 41, 49, 389, 789 evaluating, 150 factoring to find the zeros of, 55 graphing, 425 identifying, 11, 49, 389, S61 inverse, 533, S62 inverse trigonometric, 533, 534 quadratic, S65 transformations of, 838, S63 graphing functions, see Functions graphing lines, 191, 197 point-slope form, 190, 191, 194, 198, 199 proof of, 190 slope-intercept form, 188, 190, 191, 194 proof of, 196 inequalities compound, 126 graphing, linear, 249, S59 properties of, 330 solving, 805 compound, 330 linear, 26, 109, 172, 175, 176, 249, 338, |
341, 343, 345, 435, S60 systems, S68 triangle, 331 in two triangles, 340–342 writing, S59 intercepts x-intercept, 187, 191 finding, 523 identifying, 259 y-intercept, 187–189, 191 finding, 523 identifying, 259 inverse variation, 161 linear equations, see Equations linear inequalities, see Inequalities lines of best fit, 199 literal equations, 169 matrices, S69 monomials, S64 On Track for TAKS, 42, 152–153, 266, 330, 346, 389, 501, 533, 588, 713, 838 ordered pair, 11, 49, S56, see also Coordinate plane polynomial, degree of, S64 quadratic equations, see Equations radical equations, 49, S68 radicals, simplifying, 44, 519–521 rate of change, 182, see also Slope relations, S61 regression, see Lines of best fit sequences, 558 simplifying expressions, see Expressions slope(s), 182–185, 188, 322, 324, 539 finding, 279 formula, 182, 183, 185, 186, 199 of parallel lines, 184–186, 188, 306 of perpendicular lines, 184–186, 189, 306, 617 point-slope form, 303, 305 through two points, 182, 183, 185, 186, 558 of vertical lines, 182 solving equations, see Equations solving inequalities, see Inequalities systems of equations, see Equations x-intercept, see Intercepts y-intercept, see Intercepts Index S161 S161 Algebraic proof, 104–107 Alhambra, 50 Alice’s Adventures in Wonderland, Angle Bisector Theorem, 301 Converse of the, 301 Angle bisectors, 23, 300–303 102 Alternate exterior angles, 147 Alternate Exterior Angles Theorem, constructing, 23 of a triangle, 480 Angle measures, triangle classification 156 Converse of the, 163 proof of the, 164 proof of the, 159 Alternate interior angles, 147 Alternate Interior Angles Theorem, 156 Converse of the, 163 proof of the, 168 proof of the, 156 Altitude of cones, 690 of prisms or cylinders, 680 of pyramids, 689 of triangles, 314–317 Ames room, 149 Andersen, Hans Christian, 167 Angle(s), 20 acute, 21 adjacent, 28 alternate exterior, 147 alternate interior |
, 147 base, see Base angles central, see Central angles complementary, 29 congruent to a given angle, constructing, 22 corresponding, 147, 231 of depression, 544–546 of elevation, 544–546 exterior, 225, 384 exterior of an, 20 formed by parallel lines and transversals, 155–157 included, 242 inscribed, see Inscribed angles interior, 225 interior of an, 20 measure of an, 20 measuring and constructing, 20–24 naming, 20 obtuse, 21 opposite, of quadrilaterals, 391 pairs of, 28–31 reference, 570 remote interior, 225 right, 21 of rotational symmetry, 857 same-side interior, 147 straight, 21 supplementary, 29 trisecting, 25 types of, 21 vertex, 273 vertical, 30 Angle-angle-side (AAS) congruence, 254 proof of, 254 by, 216 Angle relationships in circles, 780–786 in triangles, 223–226 Angle-side-angle (ASA) congruence, 252 Animation, 53, 105, 835, 842 Annulus, 612 Answers, choosing combinations of, 816–817 Anthropology, 802 Antikythera, 792 Antique speakers, 692 Apothem, 601 Applications Advertising, 499 Agriculture, 765 Animation, 53, 835, 842 Anthropology, 802 Archaeology, 262, 787, 793 Architecture, 47, 159, 166, 220, 324, 457, 467, 485, 529, 658, 667, 695, 706, 767, 843, 859, 875 Art, 10, 32, 167, 465, 483, 557, 593, 657, 668, 834, 849, 860, 863, 873, 876 Art History, 52 Astronomy, 227, 274, 494, 720, 752, 844, 877 Aviation, 229, 277, 546, 547, 564 Bicycles, 337 Biology, 75, 77, 83, 100, 185, 604, 685, 715, 784, 857 Bird Watching, 401 Building, 538 Business, 108, 194, 312, 625 Carpentry, 18, 168, 304, 325, 408, 412, 418, 434, 555, 710, 836 Cars, 396, 425 Chemistry, 100, 683, 828, 868 City |
Planning, 305, 827 Communication, 634, 802 Community, 310 Computer Graphics, 495 Computers, 352 Conservation, 271 Consumer, 48, 684, 760 Crafts, 37, 38, 219, 357, 408, 422, 432, 594 Cycling, 538 Design, 311, 313, 317, 318, 336, 360, 403, 433 Drama, 610 Ecology, 108, 248 Electronics, 692 Engineering, 115, 233, 234, 243, 260, 412, 472, 554, 795, 841, 845 Entertainment, 149, 341, 360, 624, Angle-angle (AA) similarity, 470 625, 803, 833 S162 S162 Index Finance, 108, 522 Food, 195, 603, 656, 701–703, 718, 721 Football, 566 Forestry, 548 Games, 592, 852 Gardening, 422, 597 Geography, 39, 177, 186, 610, 626, 720, 729, 861 Geology, 86, 547, 709, 796, 804 Graphic Design, 498, 752 Health, 343 History, 48, 413, 531, 703, 778 Hobbies, 235, 464, 466, 596, 773 Home Improvement, 596 Indirect Measurement, 323 Industrial Arts, 77 Industry, 344 Interior Decorating, 609, 867 Jewelry, 719 Landscaping, 607, 686, 702 Manufacturing, 38, 754 Marine Biology, 698, 720 Math History, 25, 78, 493, 566, 768 Measurement, 404, 488, 491, 520–522, 531, 547, 596, 605 Mechanics, 434 Media, 88 Meteorology, 85, 476, 675, 703, 797, 801 Movie Rentals, 107 Music, 24, 157, 176, 218, 601 Navigation, 228, 271, 278, 402, 558, 567, 729, 767 Nutrition, 107 Oceanography, 174 Optics, 868 Optometry, 877 Orienteering, 556 Parachute, 302 Parking, 159 Pets, 361 Photography, 385, 459, 475 Physical Fitness, 79 Physics, 25, 565, 861, 867 Political Science, 79, 93 Problem-Solving Applications, 30, 105, 193, 252–253, 315 |
–316, 428, 456, 528, 618, 749, 825 Racing, 392 Real Estate, 486 Recreation, 15, 92, 108, 271, 476, 564, 636, 673, 674, 828, 850 Safety, 349, 353, 386, 395, 530 Sailing, 245 Science, 786 Shipping, 395 Shuffleboard, 305 Social Studies, 403 Space Exploration, 354, 491, 492, 751 Space Shuttle, 548 Sports, 17, 19, 40, 46, 149, 165, 175, 259, 458, 492, 530, 562, 603, 635, 720, 729, 761, 851 Surveying, 25, 224, 256, 257, 263, 276, Assessment Test Prep 353, 474, 547, 556 Technology, 92, 809 Textiles, 125 Theater, 246 Transportation, 183, 194, 360, 620, 631, 633, 866 Travel, 17, 54, 84, 335, 458, 484 Appropriate methods choosing, 372, 373, 616, 619, 620 Appropriate units choosing, 596 Approximating, 37, 335, 360, 460–461, 484, 491–492, 549, 577–579, 796 Arc intercepted, 772 major, 756 minor, 756 Arc Addition Postulate, 757 Arc length, 766 Arc marks, 22 Archaeology, 262, 787, 793 Archery, 635 Archimedes, 599, 703 Architecture, 47, 159, 166, 220, 324, 457, 467, 485, 529, 658, 667, 695, 706, 767, 843, 859, 875 Arcs, 756 adjacent, 757 chords and, 756–759 congruent, 757 measure, 756 Are You Ready?, 3, 71, 143, 213, 297, 377, 451, 515, 585, 651, 743, 821 Area(s), 36, 754, 815, 818 of circles, 37, 600 in the coordinate plane, 616–619 under curves, estimating, 621 of kites, 591 lateral, see Lateral area of lattice polygons, developing Pick’s Theorem for, 613 of parallelograms, 589 perimeter and, 36 |
proportional, 490 of regular polygons, 601 of rhombuses, 591 of sectors, 764–766 of segments, 765 of spherical triangles, 727 surface, see Surface area of trapezoids, 590 of triangles, 590 Area Addition Postulate, 589 Area ratio, 490 Argument, convincing, writing a, 379 Armstrong, Lance, 337 Arrow notation, 50 Art, 10, 32, 167, 465, 483, 577, 593, 657, 668, 834, 849, 860, 863, 873, 876 Art History, 52 Artifacts, 26 ASA (angle-side-angle) congruence, 252 Chapter Test, 64, 134, 206, 288, 370, 442, 508, 576, 644, 734, 814, 888 College Entrance Exam Practice ACT, 207, 289, 889 SAT, 65, 443, 509 SAT Mathematics Subject Tests, 135, 371, 577, 735, 815 SAT Student-Produced Responses, 645 Cumulative Assessment, 68–69, 138–139, 210–211, 292–293, 374–375, 446–447, 512–513, 580–581, 648–649, 738–739, 818–819, 892–893 Multi-Step TAKS Prep, 34, 58, 102, 126, 180, 200, 238, 280, 328, 364, 406, 436, 478, 502, 542, 568, 614, 638, 678, 724, 770, 806, 854, 880 Multi-Step TAKS Prep questions are also found in every exercise set. Some examples are: 10, 18, 26, 32, 39 Ready to Go On?, 35, 59, 103, 127, 181, 201, 239, 281, 329, 365, 407, 437, 479, 503, 543, 569, 615, 639, 679, 725, 771, 807, 855, 881 Standardized Test Prep, 69, 139, 211, 293, 375, 447, 513, 581, 649, 739, 819, 893 Study Guide: Preview, 21, 72, 144, 214, 298, 378, 452, 516, 586, 652, 744, 822 Study |
Guide: Review, 60–63, 130–133, 202–205, 284–287, 366–369, 438–441, 504–507, 572–575, 640–643, 730–733, 810–813, 884–887 TAKS Prep, 68–69, 138–139, 210–211, 292–293, 374–375, 446–447, 512–513, 580–581, 648–649, 738–739, 818–819, 892–893 TAKS Tackler Any Question Type Check with a Different Method, 372–373 Estimate, 578–579 Highlight Main Ideas, 890–891 Identify Key Words and Context Clues, 290–291 Interpret Coordinate Graphs, 208–209 Interpret a Diagram, 510–511 Measure to Solve Problems, 736–737 Use a Formula Sheet, 646–647 Gridded Response: Record Your Answer, 136–137 Multiple Choice, Eliminate Answer Choices, 444–445 Recognize Distracters, 816–817 Work Backward, 66–67 Test Prep questions are found in every exercise set. Some examples are: 11, 19, 26, 33, 40 Asterism, 227 Astronomy, 227, 274, 494, 720, 752, 844, 877 Auxiliary line, 223 Aviation, 229, 277, 546, 547, 564 Axioms, see Postulates Axis of a cone, 690 of a cylinder, 681 polar, 808 of symmetry, 362, 856 symmetry about an, 858 B Bar graph, S78 Bascule bridges, 895 Base(s) of cones, 654 of cylinders, 654 of isosceles trapezoids, 426 of isosceles triangles, 273 of prisms, 654 of pyramids, 654 of trapezoids, 429 of triangles, 36 Base angles, 273 of isosceles triangles, 273 of trapezoids, 429 Base edges, 680 Baseball fields, 43 Bathysphere, 720 Bearing of a vector, 560 Berg, Bryan, 146 Between, 12, 14 Biconditional statements, 96–98 Bicycles, 337 Big Bend National Park, 626, 678 Big Tex, 518, 520 Binomials, multiplication of, 40, 592 Biology, 75, 77, 83, 100 |
801 developing formulas for, 600–602 equations of, see Equations of circles exterior of a, 746 graphing, 800–805 great, see Great circle inscribed, 309, 313 interior of a, 746 lines that intersect, 746–750 sector of a, 764 segment of a, 765 segment relationships in, 790–795 segments that intersect, 746 tangent, 747 through three noncollinear points, constructing, 763 unit, see Unit circle Circle graphs, 26, 755, S80 Circumcenter of a triangle, 307 constructing, 307 Circumcenter Theorem, 308 proof of the, 308 Circumference, 37, 600 and area of a circle, 37 of a great circle of a sphere, 769 Circumscribe, 308 a circle about a triangle, constructing, 313, 778 an equilateral triangle about a circle, constructing, 779 Circumscribed circle, 308, 313 City Planning, 305, 827 Classifying pairs of lines, 192 triangles, 230 Clinometer, 550 Clouds, 675 CN Tower, 843 Coinciding lines, 192 Coins, 741 College Entrance Exam Practice, see also Assessment ACT, 207, 289, 889 SAT, 65, 443, 509 SAT Mathematics Subject Tests, 135, 371, 577, 735, 815 appropriate units, 596 combinations of answers, 816–817 Chord-Chord Product Theorem, 792 SAT Student-Produced Responses, 645 Collinear points, 6 Common Angles Theorem, 117 proof of the, 797 Chords, 746 arcs and, 756–759 Circle(s), 600, 742–819 angle relationships in, 780–786 area of, 37, 600 centers of, 600 constructing, 774 circumference of a, 37, 600 circumscribe a, about a triangle, 778 proof of the, 117 Common Segments Theorem, 118 Converse of the, 119 proof of the, 119 proof of the, 118 Common tangent, 748 Communication, 634, 802 compare, 98, 148, 165, 185, 245, 593, 656, 700, 717 S164 S164 Index 218, 245, 342, 359, 537, 563, 602, 608, 619, 664, 683, 841, 850, 858, 874 discuss, 624 explain, 8, 16, 24, 31, 46, |
52, 76, 90, 107, 122, 157, 165, 174, 185, 193, 218, 226, 245, 255, 262, 269, 276, 303, 310, 324, 335, 342, 352, 359, 385, 421, 431, 457, 473, 490, 497, 520, 546, 563, 593, 602, 608, 619, 633, 673, 683, 692, 700, 708, 717, 750, 766, 775, 786, 801, 826, 833, 850, 858, 866, 874 justify, 394, 801 list, 113, 148, 352 name, 8, 24, 218, 233, 593 sketch (draw), 8, 16, 24, 31, 46, 52, 218, 226, 255, 269, 276, 310, 317, 359, 385, 394, 401, 421, 464, 473, 484, 490, 528, 546, 554, 593, 750, 759, 786, 801, 850, 858 summarize, 352 write, 16, 31, 37, 46, 255, 262, 303, 324, 342, 401, 411, 421, 431, 457, 464, 473, 484, 490, 497, 520, 528, 537, 546, 619, 683, 692, 700, 750, 759, 775, 786, 801 Community, 310 Comparing surface areas and volumes, 722–723 Comparison Property of Inequality, 330 Compass, 14, F47 Compass and straightedge, see Construction(s), using compass and straightedge Complement of an event, 628 Complementary angles, 29 Complements, 29 Component form of a vector, 559 Composite figures, 606–608, 818 measuring, 611 Compositions of transformations, 848–850 Compound inequalities, 126 solving, 330 Compound statements, 128 Computer-animated films, 835 Computer Graphics, 495 Computers, 352 Concave polygons, 383 Concentric circles, 747 Conclusion, 81 Concurrency, point of, 307 Concurrent lines, 307 Conditional statements, 81–84 Conditionals, related, 83 Conditions for Parallelograms, 398, 399 proof of the, 398 Conditions for Rectangles, 418 Conditions for Rhombuses, |
419 Cones, 654 altitude of, 690 axis of, 690 double, 660 drawing, 653 frustum of, 668, 696 oblique, 690 right, see Right cones surface area of, 689–692 vertex of, 690 volume of, 705–708 Congruence properties of, 106 triangle, see Triangle congruence Congruence transformations, 824, 854 Congruent angles, 22 Congruent arcs, 757 Congruent Complements Theorem, 112 proof of the, 112 Congruent polygons, 231 properties of, 231 Congruent segments, 13 Congruent Supplements Theorem, 111 proof of the, 111 Congruent triangles, 231–233 constructing using ASA, 253 using SAS, 243 properties of, 231 Conjecture, 74, 171, 188, 189, 222, 241, 250, 251, 278 making a, 321, 331, 381, 390, 416, 417, 426, 613, 669, 676, 781, 790, 847 using deductive reasoning to verify a, 88–90 using inductive reasoning to make a, 74–76 Conjunction, 128 Conservation, 271 Constant of variation, 501, S62 Construction(s), 14, 17, 79, 177, 248, 258, 306, 313, 404, 424, 487 For a complete list, see page S87 angle bisector, 23 perpendicular lines, 179 proving valid, 282–283 segment bisector, 16 using compass and straightedge angle congruent to a given angle, 22 center of a circle, 774 centroid of a triangle, 314 circle through three noncollinear points, 763 circumcenter of a triangle, 307 circumscribe a circle about a triangle, 778 midsegment of a triangle, 327 orthocenter of a triangle, 320 parallel lines, 163, 170–171, 179 parallelogram, 404 perpendicular bisector of a segment, 172 perpendicular lines, 179 reflections, 829 regular polygons, 380–381 decagon, 381 dodecagon, 380 hexagon, 380 octagon, 380 pentagon, 381 square, 380 rhombus, 415 right triangle, 258 rotations, 844 segment congruent to a given segment, 14 segment of given length, 18 tangent to a circle at a point, 748 tangent to a circle from an exterior point, 779 translations, 836 using geometry software |
, 154, 480, 781 midpoint, 12 special points in triangles, 321 transformations, 56–57 congruent triangles, 249 similar triangles, 468–469 using patty paper midpoint, 16 parallel lines, 171 reflect a figure, 824 rotate a figure, 839 translate a figure, 831 Consumer Application, 48, 684, 760 Contraction, 873 Contradiction, proof by, 332 Contrapositive, 83 Law of, 83 Converse, 83 of a theorem, 162 Converse of the Alternate Exterior Angles Theorem, 163 proof of the, 164 Converse of the Alternate Interior Angles Theorem, 163 proof of the, 168 Converse of the Angle Bisector Theorem, 301 Converse of the Common Segments Theorem, 119 proof of the, 118 Converse of the Corresponding circumscribe an equilateral triangle Angles Postulate, 162 about a circle, 779 congruent triangles using ASA, 253 congruent triangles using SAS, 243 dilations, 872, 878 equilateral triangle, 220 incenter of a triangle, 313 irrational numbers, 363 kites, 435 line parallel to side of triangle, 481 Converse of the Hinge Theorem, 340 proof of the, 340 Converse of the Isosceles Triangle Theorem, 273 Converse of the Perpendicular Bisector Theorem, 300 Converse of the Pythagorean Theorem, 350 Converse of the Same-Side Interior Angles Theorem, 163 proof of the, 168 Converse of the Triangle Proportionality Theorem, 482 Convex polygons, 383 Convincing argument, writing a, 379 Coordinate plane, 42, 43, 361, 393, 397, 400, 402, 405, 410, 420–423, 434, 435, S56 area in the, 616–619 circles in the, 799–801 dilations in the, 495–497, 874 distance in the, 43–46 graphing in the, 42, 208–209 lines in the, 190–193 midpoint in the, 43–46 parallelograms in the, 393 perimeter in the, 616–619 reflections in the, 826 rotations in the, 840 similarity in the, 495–497 strategies for positioning figures in the, 267 transformations in the, 50–52 translations in the, 832 Coordinate proof, 267–269, 275, 313, 319, |
96 the Pythagorean Theorem, 522 Design, 311, 313, 317, 318, 336, 360, 403, 433 Detachment, Law of, 89 Diagonal, 48 of the polygon, 382 of a right rectangular prism, 671 Diagrams, 73, S40 interpreting, 510–511 Diameter, 37, 747 Dilations, 495, 872–874 center of, 872 in the coordinate plane, 495–497, 874 of figures, constructing, 878 Dimensions changing, see Effects of changing dimensions three, see Three dimensions Direct reasoning, 332 Direct variation, 161, 501, S62 Direction of a vector, 560 Disjunction, 128 Disjunctive Inference, Law of, 129 Displacement, 703 Distance, 13 solving, 25 in the coordinate plane, 43–46 between a point and a line, 301 from a point to a line, 172 Distance Formula, 44 proof of the, 354 in three dimensions, 672 Distance Function, using, 263, 271, 272 Distributive Property, 104, S51 Division Property of Equality, 104 Division Property of Inequality, 109, 330 Dodecagons, 382 regular, 380 Dodecahedron, 669 Domain, 41, 389, 405, 533, 547 Double cone, 660 Drama, 610 Drawing(s), 17 diagram that represents information, 19 isometric, 662 one- and two-point perspective, 668 orthographic, 661 perspective, 662 segments, 14 Dual of a tessellation, 868 Dulac, Edmund, 167 E Earthquakes, 804 Ecology, 108, 248 Edge base, 680 lateral, 680 of a three-dimensional figure, 654 Effects of changing dimensions, 683, 691, 700, 708, 713, 716 proportionally, 622–624 Egypt, ancient, 353 Electronics, 692 The Elements, 257 Elevation, angles of, 544–546 Elimination, solving systems of equations by, 152–153, 157, 193, S67 Endpoints, 7, 9 Engineering, 115, 233, 234, 243, 260, 412, 472, 554, 795, 841, 845 Enlargement, 495, 873 Entertainment, 149, 341, 360, 624, 625, 803, 833 Epicenter, 804 Equal vectors, 561 Equality, properties of, 104 Equation of a Circle Theorem, |
799 Equations of circles, 799 finding, 805 of a horizontal line, 190 of lines, 303–306, 308, 311–313, 315–318, 339 literal, 588, 590 quadratic, 266 linear, 11, 15–19, 22–26, 29, 31–34, 38–41, 44, 104–109, 124, 155, 156, 158, 159, 219–221, 227, 228, 230, 235–237, 245, 246, 249, 259, 264, 265, 272, 276, 277, 301, 302, 304, 305, 318, 320, 325, 337, 349, 352–355, 384–388, 392, 393, 395–397, 403, 405, 406, 409, 410, 412, 423, 425, 430, 432, 433, 434, 751, 753, 760–762, 776–779, 786–789, 795–798, 807, 814, S58 literal, 41, 169 quadratic, 27, 228, 230, 235, 237, 246, 259, 277, 326, 349, 350, 352, 355, 388, 415, 430, 432–434, 494, 752, 771, 777, 796–798, 804, 814, S66 radical, 49 systems of, 125, 158, 159, 176, 177, 194, 805 by elimination, 152–153, 157, 193, S67 by graphing, 8, 193, 195, 196 by substitution, 316–318, 396, S67 of a vertical line, 190 writing, 11, 18, 25 linear, 19, 31–34, 38–41 of lines, 190–197 literal, 41 Equiangular Triangle Corollary, 275 Equiangular triangles, 216 Equidistant, 300, 746, 799 Equilateral triangle, circumscribed about a circle, 779 Equilateral Triangle Corollary, 274 Equilateral triangles, 217, 273–276 constructing, 220 Error Analysis, 18, 39, 92, 124, 160, 195, 236, 258, 325, 353, 387, 413, 423, 476, 499, 522, 540, 557, 604, 634, 659, 711, 762, 797, 852, 877 Escher, M. C., 861, 868, 876, 880 Est |
imating area under curves, 621 Estimation, 25, 37, 41, 77, 108, 177, 195, 229, 278, 325, 361, 387, 433, 466, 492, 493, 538, 565, 611, 621, 676, 719, 768, 803, 844, 877, S52 rounding and, S52 Estimation strategies, 578–579 Euclid, 257, 460 Euler, Leonhard, 78 Euler line, 321 Euler’s Formula, 670 Event, 628 complement of an, 628 Exam, final, preparing for, 823 Expansion, 873 Experiment, 628 fair, 628 Experimental probability, 798 Exponents, S53 properties of, S54 S166 S166 Index Expressions, 3 evaluating, 19, 162, 164–167, 334, 336, 384, 392, 393, 395, 399, 402, 405, 408–410, 412, 429, 432, 433, S57 simplifying, 19, 36, S50 writing, 229, 788, S57 Extended Response, 69, 139, 178, 211, 230, 259, 293, 355, 375, 425, 447, 487, 513, 581, 621, 649, 739, 819, 893 write extended responses, 290–291 Extension Introduction to Symbolic Logic, 128–129 Polar Coordinates, 808–809 Proving Constructions Valid, 282–283 Spherical Geometry, 726–727 Trigonometry and the Unit Circle, 570–571 Using Patterns to Generate Fractals, 882 Exterior, 225 of an angle, 20 of a circle, 746 Exterior Angle Theorem, 225 Exterior angles, 225, 384 Exterior point, constructing tangent to a circle from an, 779 External secant segment, 793 Extremes, 455 F Face lateral, 680 of a three-dimensional figure, 654 Factoring to find the zeros of each function, 55 solving by, 388, S66 using, 279 Fahrenheit (F) degrees, 105 Fair experiment, 628 Fair Park, 803 Fiber-optic cables, 28 Fibonacci sequence, 78, 461 Figures composite, see Composite figures three-dimensional, see Three- dimensional figures Final exam, preparing for your, 823 Finance, 108, 522 Finding slope, see Slope Fitness Link, 539 Flash cards, |
587 Flatiron Building, 220 Flips, see Reflections Flowchart proofs, see Proofs, flowchart FOIL, 592 Food, 195, 603, 656, 701–703, 718, 721 Football, 566 Forbidden City, 48 Forestry, 548 Formula sheet, using a, 646–647 Formulas, see back cover deriving, 39, 541, 696 developing, 589–591, 600, 601 for circles, 600 for regular polygons, 601 for triangles, 590 for quadrilaterals, 589–591 functional relationships in, 713, S63 memorize, 587 in three dimensions, 670–673 using, 36–37 45°-45°-90° triangle, 356 Four Chromatic Gates, 650 Fractals, 882 Freescale Marathon, 140 Fresnel, Augustine, 894 Fresnel lenses, 894 Frieze pattern, 863 Frustum of a cone, 668, 696 of a pyramid, 696 Functional relationships in formulas, 713, S63 Functions, 11, 41, 49, 389, 789 evaluating, 150 factoring to find the zeros of, 55 graphing, 425 identifying, 11, 49, 389, S61 inverse, 533, S62 inverse trigonometric, 533, 534 quadratic, S65 transformations of, 838, S63 G Games, 592, 852 Gardening, 422, 597 Garfield, James, 595 Gears, 845 General translations in the coordinate plane, 832 Geodesic dome, 234 Geography, 39, 177, 186, 610, 626, 720, 729, 861 Geology, 86, 547, 709, 796, 804 Geometric mean, 819 Geometric Means Corollaries, 519, 540 Geometric probability, 630–633 using, to estimate π, 637 Geometric proof, 110–113 Geometry hyperbolic, 729 non-Euclidean, 726 spherical, see Spherical geometry using formulas in, 36–37 Geometry Lab, see also Technology lab Construct Parallel Lines, 170–171 Construct Perpendicular Lines, 179 Construct Regular Polygons, 380–381 Design Plans for Proofs, 117 Develop π, 598–599 Develop Pick’s Theorem for Area of Lattice Polygons, 613 Develop the Triangle Sum Theorem, |
222 Explore Properties of Parallelograms, 390 Explore SSS and SAS Triangle Congruence, 240–241 Explore Triangle Inequalities, 331 Graph Irrational Numbers, 363 Hands-on Proof of the Pythagorean Theorem, 347 Indirect Measurement Using Trigonometry, 550 Model Right and Oblique Cylinders, 688 Solve Logic Puzzles, 94–95 Use Geometric Probability to Estimate π, 637 Use Nets to Create Polyhedrons, 669 Use Transformations to Extend Tessellations, 870–871 Geometry software, 12, 27, 56–57, 154, 230, 416–417, 668, 879, F47, see also Construction(s) Geometry symbols, reading, 215 Get Organized, see Graphic organizers Given statement, 111 Glide reflection, 848, 851 Glide reflection symmetry, 863 Global Positioning System, 556 Glossary, S115–S160 go.hrw.com, see Online Resources Goldbach, 78 Golden Ratio, 460–461 Graphic Design, 498, 752 Graphic organizers Graphic Organizers are available for every lesson. Some examples are: 8, 16, 24, 31, 37 Graphing in the coordinate plane, 42 irrational numbers, 363 Graphing calculator, 188–189, 196, 199, 761, 800, 846–847, 879 Graphing functions, see Functions Graphing lines, 191, 197, 259 point-slope form, 190, 191, 194, 198, 199 proof of, 190, 196 slope-intercept form, 188, 190, 191, 194 proof of, 196 Graphs, bar, S78 circle, 755 histogram, S78 misleading, S81 Gravity, center of, 314 Great circle, 714, 726 of a sphere, circumference of a, 769 Great Texas Balloon Race, 295 Gridded Response, 11, 69, 93, 109, 136–137, 139, 169, 211, 221, 279, 293, 313, 355, 362, 373, 375, 397, 435, 447, 477, 500, 510, 511, 513, 540, 567, 578, 579, 581, 597, 627, 647, 649, 687, 712, 739, 763, 769, 789, 819, 845, 878, 890 |
, 891, 893 Record Your Answer, 136–137 Index S167S167 H Hands-on proof of the Pythagorean Theorem, 347 Hayes, Joanna, 19 Head-to-tail vector addition method, 561 Health, 343 Height of triangle, 36 Helpful Hint, 6, 43, 83, 98, 105, 110, 112, 119, 146, 147, 156, 226, 231, 232, 261, 307, 316, 332, 334, 391, 400, 401, 410, 464, 483, 488, 495, 520, 535, 546, 551, 553, 554, 601, 623, 631, 663, 749, 764, 784, 800, 840, 863, 872 Hemisphere, 714 Henry VIII, 305 Heptagon, 382 Hexagon, 382 regular, 380, 819 Highlighting main ideas, 890–891 Hinge Theorem, 340 Converse of the, 340 proof of the, 340 Hippocrates, 611 Histogram, 677, S78 History, 48, 413, 531, 566, 595, 703, 778 math, see Math History HL (hypotenuse-leg) congruence, 255 Hobbies, 235, 464, 466, 596, 773 Hogan, 803 Home Improvement, 596 Homework Help Online Homework Help Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 24, 31, 38 Horizon, 662 Horizontal line, equation of a, 190 Horizontal translations in the coordinate plane, 832 of parent functions, 838 Hot Tip, 65, 135, 207, 289, 371, 443, 509, 577, 645, 735, 815, 889 How to Study Geometry, xx Hurricanes, 476 Hypatia, 768 Hyperbolic geometry, 729 Hypotenuse, 45 Hypotenuse-leg (HL) congruence, 255 Hypothesis, 81 I Icosahedron, 669 Identity, Pythagorean, 531–532 Image, 50 Incenter of a triangle, 309 Incenter Theorem, 309 S168 S168 Index Included angles, 242 Included sides, 252 Indirect measurement, 323, 488 using trigonometry, 550 |
Indirect proof, 332–335, 339 Inductive reasoning, 74, 75 using, to make conjectures, 74–76 Industrial Arts, 77 Industry, 344 Inequalities compound, 126 graphing, linear, 249, S59 properties of, 330 solving, 805 compound, 330 linear, 26, 109, 172, 175, 176, 249, 338, 341, 343, 345, 435, S60 systems, S68 triangle, exploring, 331 in two triangles, 340–342 writing, S59 Information not enough, 247, 248, 250, 405, 420, 422, 423, 425, 437, 440, 442, 446, 473, 512, 554, 556 too much, 209 Inscribed Angle Theorem, 772 proof of the, 772, 778 Inscribed angles, 772–775 Inscribed circle, 309, 313 Inscribed polygons, 380 Integers, 80, S53 Intercepted arc, 772 Intercepts x-intercept, 187, 191 finding, 523 identifying, 259 y-intercept, 187–189, 191 finding, 523 identifying, 259 Interior, 225 of an angle, 20 of a circle, 746 Interior angles, 225 Interior Decorating, 609, 867 Interpreting diagrams, 510–511 Intersecting lines, 192 Intersections, 8 of lines and planes, 8 Inverse, 83 Inverse functions, 533, S62 Inverse trigonometric functions, 533, 534 Inverse variation, 161 Irrational numbers, 37, 80, S53 graphing, 363 Irregular polygons, 382 Isle of Man, 861 Isometric drawings, 662 Isometry, 824 Isosceles trapezoids, 426, 429 bases of, 426 legs of, 426 Isosceles Triangle Theorem, 273 Converse of the, 273 proof of the, 273 Isosceles triangles, 217, 273–276 legs of, 273 Iteration, 882 J Jewelry, 719 Johnson, Lyndon B., 583 Johnson Space Center, 548 K Kaleidoscope, 868 Kite, 427, 433 area of, 591 constructing, 435 properties of, 427–429 proof of, 427 Know It Note Know-It Notes are found throughout this book. Some examples: 6, 7, 8, 13, 14 Koch snowflake fractal, 882 L Lady Bird Johnson Wildflower Center, 607 Land Development, |
, 302, 304, 306, 600, 714, 743, 804 Logic puzzles, solving, 94–95 Logically equivalent statements, 83 Lune, 611 Lunette, 767 Luxor Hotel, 159 M Madurodam, 458 Magnitude of a vector, 560 Main ideas, highlighting, 890–891 Major arc, 756 Make a Conjecture, 321, 331, 381, 390, 416, 417, 426, 613, 669, 676, 781, 790, 847, see also Conjecture Manufacturing, 38, 754 Mapping, 50 Marine Biology, 698, 720 Math Builders, xxiii–xxvii Math History, 25, 41, 78, 257, 318, 493, 566, 611, 703, 768 Math vocabulary, learning, 299 Matrices operations, S69 point, 846 transformations with, 846–847 McDonald Observatory, 234 Mean, 11, 43, S76 geometric, 819 Means, 455 Measurement, 404, 488, 491, 520–522, 531, 547, 585, 596, 599, 605, 611 back cover absolute error, S73 indirect, see Indirect measurement accuracy, precision, and tolerance, S72 choose appropriate units, S74 customary system of, S70, back cover metric system of, 3, S70, back cover nonstandard units, S74 relative error, S73 rates, S70 significant digits, S73 tools of, choose appropriate, S75 units, S70–S71 Measures of central tendency, 477, S76 Measuring to solve problems, 736–737 Mechanics, 434 Media, 88 Median(s), 11, S76 of triangles, 314–317 Memorize formulas, 587 Meteorology, 85, 476, 675, 703, 797, 801 Meters, 36 Metric system of measurement, 3, S70, back cover Metronome, 24 Midpoint, 12, 15 constructing, 16 in the coordinate plane, 43–46 Midpoint Formula, 43 in three dimensions, 672 Midsegment triangle, 322 Midsegments of trapezoids, 431 of triangles, constructing, 327 Migration patterns, 74 Minor arc, 756 Minute Maid Park, 43 Minutes (in degrees), 27 Möbius, August Ferdinand, 566 Mode, 11, 345, S76 Model make a, |
S41 probability, 630 Modeling oblique cylinders, 688 right cylinders, 688 Mohs’ scale, 86 Monument Link, 466 Mosaic, 376, 876 Motions, rigid, 824 Moveable bridges, 895 Movie Rentals, 107 Multi-Step Multi-Step questions appear in every exercise set. Some examples: 11, 17, 24, 25, 26 Multi-Step TAKS Prep, 34, 58, 102, 126, 180, 200, 238, 280, 328, 364, 406, 436, 478, 502, 542, 568, 614, 638, 678, 724, 770, 806, 854, 880, see also Assessment Multi-Step TAKS Prep questions are also found in every exercise set. Some examples are: 10, 18, 26, 32, 39 Multiple Choice, 66–69, 138–139, 210–211, 372–374, 444–447, 510–513, 578–581, 646–649, 736, 738–739, 816–819, 891–893 Choose Combinations of Answers, 816–817 Eliminate Answer Choices, 444–445 Work Backward, 66–67 Multiple Representations, 6, 7, 21, 50, 80–81, 83, 128, 173, 226, 255, 330, 350, 429, 455, 462, 525, 528, 561, 630, 669, 681, 690, 746–748, 756, 764–766, 785 Multiplication of binomials, 40, 592 scalar, 566 Multiplication Property of Equality, 104 Multiplication Property of Inequality, 109, 330 Municipal Marina, 142 Music, 24, 157, 176, 218, 601 Musical triangles, 218 Mystery spots, 150, 180 N n-gons, 382 Naming angles, 20 Natural numbers, 41, 80, S53 Navigation, 228, 271, 278, 402, 558, 567, 729, 767 Negation, 82 of a vector, 566 Nets, 655, 669 Network, 95 New Madrid earthquake, 804 Non-Euclidean geometry, 726 Nonagons, 382 Index S169S169 Noncollinear points, 6 three, constructing circle through, 763 Noncoplanar points, 6 Normal curve, standard, 860 |
Not enough information, 247, 248, 250, 405, 420, 422, 423, 425, 437, 440, 442, 446, 473, 512, 554, 556 Note taking Strategies, see Reading and Writing Math Number Theory, On Track for TAKS, 80 Numbers classifying, S53 estimating, S52 irrational, see Irrational numbers natural, 41, 80, S53 properties of, S51 rational, 80, S53 real, S50, S53 rounding, S52 whole, 80, S53 Numerator, 451 Nutrition, 107 O Oblique cones, 690 Oblique cylinders, 681 modeling, 688 Oblique prism, 680 Obtuse angles, 21 Obtuse triangles, 216 Oceanography, 174 Octagon, 382 regular, 380 Octahedron, 669 Olympic Games, 2004, 19 Olympus Mons, 752 On Track for TAKS Parent Resources Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 24, 31, 38 TAKS Practice Online, 68, 138, 210, 292, 374, 446, 512, 580, 648, 738, 818, 892 Op art, 860 Opposite angles of quadrilaterals, 391 Opposite rays, 7 Opposite reciprocals, 184 Opposite sides of quadrilaterals, 391 Optics, 868 Optometry, 877 Order of operations, 3, S50 of rotational symmetry, 857 Ordered pair, 11, 49, S56, see also Coordinate plane Ordered triples, 671 Orienteering, 252, 556 Origami, 238, 594 Origin, 42, S56 Orthocenter of a triangle, 316 constructing, 320 Orthographic drawings, 661 Outcome, 628 P Pairs of angles, 28 of circles, 747 of lines, 192 Algebra, 42, 152–153, 266, 330, 346, classifying, 192 389, 501, 533, 588, 713, 838 Data Analysis, 198–199, 755 Number Theory, 80 Probability, 628–629 One-point perspective, 662 drawing figures in, 668 One-to-one correspondence, 20 Online Resources Career Resources Online, 87, 237, 320, 494, 612, 805 Chapter Project Online, 2, 70, 142 |
, 212, 296, 376, 450, 514, 584, 650, 742, 820 Homework Help Online Homework Help Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 24, 31, 38 Lab Resources Online, 56, 154, 188, 250, 321, 426, 460, 468, 480, 524, 780, 790, 846 Parent Resources Online Parabola, S65 Parachute, 302 Paragraph proofs, see Proofs, paragraph Parallel lines, 142–211 constructing, 163, 170–171, 179 defined, 146 exploring, 154, 188–189 proving, 162–165 slopes of, 184–186, 188, 192, 306 and transversals, 155–157 Parallel Lines Theorem, 184 Parallel planes, 146 Parallel Postulate, 163 Parallel rays, 146 Parallel segments, 146 Parallel vectors, 561 Parallelogram lift, 396 Parallelogram mount, 398 Parallelogram vector addition method, 561 Parallelograms, 390 area of, 589 conditions for, 398–401 constructing, 404 properties of, 390–394 special, see Special parallelograms Parallels, adjustable, 836 Parent functions, 221 horizontal translations of, 838 reflections of, 838 transformations of, 838 vertical translations of, 838 Parent Resources Online Parent Resources Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 24, 31, 38 Parking, 159 Pascal’s triangle, 883 Pasteur, Louis, 828 Patterns, 327, S44 frieze, 863 looking for, 763 using, to generate fractals, 882 Patty paper, see Construction(s), using patty paper Peaucellier cell, 434 Pendulum, 25 Penny-farthing bicycle, 768 Pentagon building, 859 Pentagon, 382 regular, 381 Percent grade, 534, 536, 539, 540 Percents, 27, 41 Perimeter, 36 in the coordinate plane, 616–619 proportional, 490 ratio, 490 Perpendicular Bisector Theorem, 300 Converse of the, 300 proof of the, 300 Perpendicular bisector, 300–303 of a segment, 172 constructing, 172 Perpendicular lines, 142–211 constructing, 179 defined, 146 |
exploring, 188–189 proving, 173 slopes of, 184–186, 189, 306, 617 Perpendicular Lines Theorem, 184 Perpendicular rays, 146 Perpendicular segments, 146 Perpendicular Transversal Theorem, 173 proof of the, 173 Perspective, 481 Perspective drawings, 662 Pets, 361 pH, 96, 761, S74 Photography, 385, 459, 475 Physical Fitness, 79 Physics, 25, 565, 861, 867 Pi (π), 37 developing, 598–599 using geometric probability to estimate, 637 Pi (π) calculator key, 601 Piano strings, 155 S170 S170 Index Pick’s Theorem, 613 for area of lattice polygons, developing, 613 Pizza, 195 Plane symmetry, 858 Planes, 6, 7 intersection of lines and, 8 Platonic solids, 669 Plumb bob, 168 Point(s), 6, 7, 12 collinear, 6 of concurrency, 307 constructing a tangent to a circle at a, 748 coplanar, 6 equidistant, 300, 746, 799 exterior, constructing a tangent to a circle from an, 779 and a line, distance between a, 172, 301 noncollinear, 6 noncoplanar, 6 special, in triangles, 321 of tangency, 746 three noncollinear, constructing a circle through, 763 two, slope of a line through, 558 vanishing, 662 Point Isabel Lighthouse, 894 Point matrix, 846 Point-slope form, 190, 191, 194, 198, 199, 303, 305 proof of, 190 Pointillism, 10 Polar axis, 808 Polar coordinate system, 808–809 Polar coordinates, 808–809 Polaris, 844 Pole, 808 Political Science, 79, 93 Polygon(s), 98 concave, 383 congruent, 231 convex, 383 diagonal of the, 382 inscribing, 380 irregular, 382 lattice, 613 properties and attributes of, 382–385 quadrilaterals and, 376–449 regular, see Regular polygons sides of, 382 similar, 462–464 vertex of the, 382 Polygon Angle Sum Theorem, 383 Polygon Exterior Angle Sum Theorem, 384 Polyhedrons, 669, 670 creating, by using nets, 669 regular, 669 Polymer, 868 Pompeii, 413 Positioning |
figures in the coordinate plane, strategies for, 267 Postulates, 7 For a complete list, see pages S82–S87 Precision, 596, S72 Predicting, 634 conditions for special parallelograms, 416–417 other triangle congruence relationships, 250–251 triangle similarity relationships, 468–469 Preimage, 50 Preparing for your final exam, 823 Preparing for TAKS, TX2–TX3 Prime number, 78 Primes, 50 Prisms, 654 altitude of, 680 drawing, 653 lateral area of, 680 oblique, 680 right, see Right prisms right rectangular, diagonals of, 671 surface area of, 680–683 volume of, 697–700 Probability, 10, 32, 85, 230, 237, 339, 459, 467, 565, 628–629, 702, 769, 798, 835, S77 experimental, 798 geometric, see Geometric probability model, 630 On Track for TAKS, 628–629 theoretical, 628 Problem-Solving Applications, 30, 105, 193, 252–253, 315–316, 428, 456, 528, 618, 749, 825 Problem-Solving Handbook, S40–S49, see also Problem-Solving Strategies Problem Solving on Location, xxix Cavanaugh Flight Museum, Addison, 294 The Freescale Marathon, Austin, 140 The Great Texas Balloon Race, Longview, Estes, 295 Lyndon B. Johnson’s Birthplace, Johnson City, 583 Moveable Bridges, Quintana Island, Tule Lake, Rio Hondo, Corpus Christi, 895 Point Isabel Lighthouse, Port Isabel, 894 Reliant Stadium, Houston, 740 Reunion Tower, Dallas, 582 Show Caves, Texas Hill Country, Sonora, 141 Southwestern University, Georgetown, 448 Texas Coins, 741 Titan, Arlington, 449 Problem-Solving Plan, xxviii Problem-Solving Strategies, S40–S49 Draw a Diagram, S40 Find a Pattern, S44 Guess and Test, S42 Make a Model, S41 Make a Table, S48 Make an Organized List, S47 Solve a Simpler Problem, S49 Use a Venn Diagram, S45 Use Logical Reasoning, S46 Work Backward, S43 Problems measuring to solve, 736–737 reading |
to solve, 745 solving simpler, S49 Proof, 228, 312, 338, 391, 397, 404, 405, 411–415, 425, 427, 434, 523, 753, 758, 762, 778–779, 783, 788–789 of angle-angle-side (AAS) congruence, 254 of the Chord-Chord Product Theorem, 797 of the Circumcenter Theorem, 308 by contradiction, 332 of the Converse of the Hinge Theorem, 340 coordinate, see Coordinate proof of the Distance Formula, 354 hands-on, of the Pythagorean Theorem, 347 indirect, see Indirect proof of the Inscribed Angle Theorem, 772, 778 of the Isosceles Triangle Theorem, 273 of the Law of Cosines, 557 of the Perpendicular Bisector Theorem, 300 of the Pythagorean Theorem, 347–348, 798 of the Secant-Secant Product Theorem, 793 of the Secant-Tangent Product Theorem, 797 of the Triangle Inequality Theorem, 338 of the Triangle Midsegment Theorem, 326 of the Triangle Sum Theorem, 222, 223 The Proof Process, 112 Proofs, 104, 178 algebraic, 104–107 design plans for, 117 flowchart, 118–119, 122, 123, 168 of the Common Segments Theorem, 118 of the Converse of the Alternate Interior Angles Theorem, 168 of the Converse of the Common Segments Theorem, 118 geometric, 110–113 paragraph, 120–124, 159, 168, 169, 173, 187, 305, 306, 345, 354, 362, 396, 404, 424, 467 of the Converse of the Same-Side Interior Angles Theorem, 168 of the Perpendicular Transversal Theorem, 173 of the Same-Side Interior Angles Theorem, 159 of the Vertical Angles Theorem, 120 Index S171S171 point-slope form of a line, 190 two-column, 111–125, 159, 164, 173, 175, 187, 305, 306, 318, 342, 343, 344, 394, 395, 396, 403, 413, 423, 424, 435, 472 of the Alternate Exterior Angles Theorem, 159 of the Common Segments Theorem, 118 of the Congruent Comple |
ments Theorem, 112 of the Congruent Supplements Theorem, 111 of the Linear Pair Theorem, 111 of the Right Angle Congruence Theorem, 112 of the Vertical Angles Theorem, 120 Properties of congruence, 106 of equality, 104 of exponents, S54 of inequality, 330 of kites, 427 of linear inequalities, S60 of parallelograms, 391, 392 of real numbers, S51 of rectangles, 408 of rhombuses, 409 of squares, 410 of trapezoids, 429–431 Proportion(s), 454–457 properties of, 455 solving, 313 Proportional Perimeters and Areas Theorem, 490 Proportional relationships, 488–490 Protractor, 20, F47 using, 21 Protractor Postulate, 20 Prove statement, 111 Pyramid Arena, 695 Pyramid of Cheops, 531 Pyramids, 654 altitude of, 689 drawing, 653 frustum of, 696 regular, see Regular pyramids surface area of, 689–692 vertex of, 689 volume of, 705–708 Pythagorean Identity, 531–532 Pythagorean Inequalities Theorem, 351 Pythagorean Theorem, 45, 220, 348–352, 461, 522, 707, 749 Converse of the, 350 deriving the, 522 hands-on proof of the, 347 Math Builders, xxvi–xxvii proof of the, 347–348, 798 solving quadratic equations using, 760, 761, 770 using, in three dimensions, 671 Pythagorean triple, 349 Q Quadrants, 42, S56 Quadratic equations, see Equations Quadratic formula, using, 272, 279, S66 Quadrilaterals, 98, 382 developing formulas for, 589–593 opposite angles, 391 opposite sides, 391 polygons and, 376–449 regular, 380 special, 391 Quartiles, S80 Question type, any check with a different method, 372–373 estimate, 578–579 highlight main ideas, 890–891 interpret a diagram, 510–511 measure to solve problems, 736–737 use a formula sheet, 646–647 R Racing, 392 Radicals, simplifying, 44, 346, 519–521 Radius, 37, 747 of a sphere, 714 Rainforest Pyramid, 705, 706 Range, |
41, 345, 389, 533, 597 Rate of change, 182, see also Slope Ratio(s), 33, 454–457, 754 area, 490 perimeter, 490 rate, S70 in similar polygons, 462–464 similarity, 463, 490 trigonometric, 524, 525–528 Rational numbers, 80, S53 Rattler, 233 Rays, 7 Reading and Writing Math, 5, 73, 145, 215, 299, 397, 453, 517, 587, 653, 745, 823, see also Reading Strategies; Study Strategies; Writing Strategies Reading Math, 273, 300, 455, 456, 534, 570, 670, 748 Reading Strategies, see also Reading and Writing Math Learn Math Vocabulary, 299 Read and Interpret a Diagram, 73 Read and Understand the Problem, 453 Read Geometry Symbols, 215 Read to Solve Problems, 745 Read to Understand, 517 Use Your Book for Success, 5 Ready to Go On?, 35, 59, 103, 127, 181, 201, 239, 281, 329, 365, 407, 437, 479, 503, 543, 569, 615, 639, 679, 725, 771, 807, 855, 881, see also Assessment Real Estate, 486 Real numbers classifying, S53 operations with, S50 Reasonableness, 66–67, 332, 428, 445, 453, 578–579, 632, 749, S52 Reasoning deductive, see Deductive reasoning direct, 332 inductive, see Inductive reasoning logical, S46 spatial, 650–741 Reciprocals, opposite, 184 Recreation, 15, 92, 108, 271, 476, 564, 636, 673–674, 828, 850 Rectangle, 36 properties of, 408 proof of, 408 Rectangular prism, right, diagonal of a, 671 Reduction, 495, 873 Reference angle, 570 Reflection symmetry, glide, 863 Reflections, 50, 824–826 constructing, 829 in the coordinate plane, 826 describing transformations in terms of, 850 of figures, constructing, 824 glide, 848, 851 of parent functions, 838 Reflexive Property, 168, 176 of congruence, 106 of equality, 104 of similarity, 473 Regression, 494, see also Lines |
of best fit Regular polygons, 380–382, 818–819 area of, 601 center of, 601 central angles of, 601 constructing, 380–381 developing formulas for, 600–602 Regular polyhedrons, 669 Regular pyramids, 689 lateral area of, 689 slant height of, 689 surface area of, 689 Regular tessellations, 864 Related conditionals, 83 Relations, 389, S61 Relationships functional, in formulas, 713, S63 proportional, 488–490 Relative error, S73 Reliant Stadium, 140 Remember!, 36, 82, 104, 106, 129, 182, 191, 217, 242, 260, 269, 275, 282, 283, 309, 348, 351, 358, 382, 383, 384, 393, 398, 420, 429, 454, 489, 552, 560, 562, 589, 590, 592, 602, 617, 630, 682, 698, 765, 801, 831, 841 Remote interior angles, 225 Repeat unit, 868 Representations of three- dimensional figures, 661–664 S172 S172 Index Resultant vectors, 561 Reunion Tower, 2, 582 Rhombus(es), 409 area of, 591 conditions for, 419 constructing, 415 properties, 409 proof, 409 Rhombus method, 170 Richter scale, S74 Right angle, 21 Right Angle Congruence Theorem, 112 proof of the, 112 Right cone, 690 lateral area of, 690 slant height of, 690 surface area of, 690 Right cylinder, 681 lateral area of, 681 modeling, 688 surface area of, 681 Right prism, 680 lateral area of, 680 surface area of, 680 Right rectangular prism, diagonal of a, 671 Right triangle(s), 216 constructing, 258 similarity in, 518–520 solving, 534–537 special, 526, 529, 530 trigonometry and, 512–583 Rigid motions, 824 Rigidity, triangle, 242 Rio Grande river, 186 Rise, 182 Roberval, Gilles Personne de, 404 Roller coasters, 92, 233, 449 Roman numerals, 42 Rotational symmetry, 857 angle of, 857 order of, 857 Rotations, 50, 74, 839–841 constructing, 844 in the coordinate plane, 840 of figures, |
constructing, 839 Ruler, F47 Ruler Postulate, 13 Run, 182 S 7A Ranch, 70 Safety, 158, 349, 353, 386, 395, 530 Sailing, 245 Salinon, 768 Same-side interior angles, 147 Same-Side Interior Angles Theorem, 156 Converse of the, 163 proof of the, 168 proof of the, 159 Sample space, 628 San Jacinto Monument, 514 SAS (side-angle-side) congruence, 243 applying, 242–245 exploring, 240–241 SAS (side-angle-side) similarity, 471 Satellite, 797 Scalar multiplication, 566 Scale, 489 Scale drawing, 489 Scale factor, 495, 872 Scalene triangle(s), 217 constructing, 248, 313 Scatter plots, 198, S79 Scavenger Hunt, xxii Science, 92, 786 Scientific notation, S54 Scoring Rubric, 208 Secant, 531, 746 Secant-Secant Product Theorem, 793 proof of the, 793 Secant segment, 793 Secant-Tangent Product Theorem, 794 proof of the, 797 Seconds (in degrees), 27 Sector of a circle, 764 area of, 764–766 Segment(s), 7 of a circle, 765 area of a, 765 congruent to a given segment, 14 constructing, 14 of given length, 18 constructing, 18 measuring and constructing, 13–16 secant, 793 tangent, 794 that intersect circles, 746 Segment Addition Postulate, 14 Segment bisectors, 16 constructing, 16 Segment relationships in circles, 790–795 Seismograph, 804 Selected Answers, S88–S114 Self-similar figures, 882 Semicircle, 756 Semiregular tessellations, 864 Sequences, 558 Shen Kua, 493 Shipping, 395 Short Response, 26, 69, 101, 139, 161, 211, 293, 306, 345, 375, 405, 414, 447, 459, 467, 511, 513, 549, 579, 581, 636, 649, 667, 739, 798, 819, 878, 893 Write Short Responses, 208–209 Shuffleboard, 305 Shuffleboard Link, 305 Side-angle-side (SAS) congruence |
, 243 Side-angle-side (SAS) similarity, 471 Side lengths, triangle classification by, 217 Side-side-side (SSS) congruence, 242 Side-side-side (SSS) similarity, 470 Sides corresponding, 231 opposite, of quadrilaterals, 391 of polygons, 382 of triangles, included, 252 Sierpinski tetrahedron, 883 Sierpinski triangle, 882 Significant digits, S73 Similar, 462 Similar polygons defined, 462 ratios in, 462–464 Similar triangles angle-angle (AA), 470 applying properties of, 481–484 in the coordinate plane, 495–497 properties of, 473 in right triangles, 518–520 side-angle-side (SAS), 471 side-side-side (SSS), 470 Similarity ratio, 463, 490 Similarity transformations, 873 Simplest radical form, 346 Simplifying expressions, see Expressions Sine, 525, 841 Sines, Law of, 551–554 ambiguous case of the, 556 Sketch, 14, 17 Skew lines, 146 Skew rays, 146 Skew segments, 146 Skills Bank, S50–S81 Slant height, 818 of regular pyramids, 689 of right cones, 690 Slides, see Translations Slope(s), 182–185, 188, 322, 324, 539 formula, 182, 183, 185, 186, 199 finding, 279 through two points, 182, 183, 185, 186, 558 of parallel lines, 184–186, 188, 306 of perpendicular lines, 184–186, 189, 306, 617 point-slope form, 303, 305 of vertical lines, 182 Slope-intercept form, 188, 190, 191, 194 proof of, 196 Social Studies, 403 Solids, 654, see also Three-dimensional figures Platonic, 669 Solving compound inequalities, 330 equations, see Equations inequalities, see Inequalities Southwestern University, 448 Space, 671 Space Exploration, 354, 491, 492, 751 Space Shuttle, 548 Index S173S173 Special parallelograms, 410 conditions for, 416–421 properties of, 408–411 Special points in triangles, 321 Special quadrilaterals, 391 Sphere, 714–717, 805, 819 center of a, 714 circumference of a great circle of |
a, 769 defined, 714 drawing, 653 radius of a, 714 surface area of a, 716 volume of a, 769 Spherical geometry, 726–729 Spherical Geometry Parallel Postulate, 726 Spherical Triangle Sum Theorem, 726 Spherical triangles, area of, 727 Sports, 17, 19, 40, 46, 149, 165, 175, 259, 458, 492, 530, 562, 603, 635, 720, 729, 761, 851 Spreadsheet, 541 Springboks, 271 Square, 36, 410 properties of, 410 proof of, 410 Square base in origami, 594 Square roots, S55 simplifying, S55 Square units, 36 Square window on calculator screen, 189 SSS (side-side-side) congruence, 242 applying, 242–245 exploring, 240–241 SSS (side-side-side) similarity, 470 Standard normal curve, 860 Standardized Test Prep, 69, 139, 211, 293, 375, 447, 513, 581, 649, 739, 819, 893, see also Assessment Statements biconditional, 96–98 compound, 128 conditional, 81–84 logically equivalent, 83 Statue of Liberty, 466 Stonehenge, 787 Straight angles, 21 Straightedge, 14, F47, see also Construction(s), using compass and straightedge Strategies for positioning figures in the coordinate plane, 267 Student to Student, 21, 121, 157, 233, 359, 411, 463, 535, 632, 662, 800, 865 Study Guide: Preview, see Assessment Study Guide: Review, see Assessment Study Strategies, see also Reading and Writing Math Memorize Formulas, 587 Prepare for Your Final Exam, 822 Take Effective Notes, 145 Substitution, solving systems of equations by, 316–318, 396, S67 Substitution Property of Equality, 104 Subtend, 772 Subtraction Property of Equality, 104 Subtraction Property of Inequality, 330 Summarizing, 791 Supplementary angles, 29 Supplements, 29 Surface, lateral, of a cylinder, 681 Surface area of cones, 689–692 of cubes, 680 of cylinders, 680–683 of prisms, 680–683 of pyramids, 689–692 of regular pyramids, 689 of right cones, 690 of right cylinders, 681 of right |
prisms, 680 of spheres, 716 of three-dimensional figures, 680 and volume, comparing, 722–723 Surveying, 25, 224, 256, 257, 263, 276, 353, 474, 547, 556 Swing bridges, 895 Syllogism, Law of, 89 Symbolic logic, 128–129 Symbols, see back cover Symmetric Property, 168, 176 Symmetric Property of Congruence, 106 Symmetric Property of Equality, 104 Symmetric Property of Similarity, 473 Symmetry, 856–858 about an axis, 858 axis of, 362, 856, S65 defined, 856 glide reflection, 863 identifying, in three dimensions, 858 line, 856 line of, 318, 856 plane, 858 rotational, see Rotational symmetry translation, 863 Systems of equations, see Equations T Tables making, 230, 763, S48, S78 using, 19, 40 Taffrail log, 278 TAKS Practice, S4–S39 TAKS Prep, 68–69, 138–139, 210–211, 292–293, 374–375, 446–447, 512–513, 580–581, 648–649, 738–739, 818–819, 892–893, see also Assessment TAKS Tackler, see also Assessment Any Question Type Check with a Different Method, 372–373 Estimate, 578–579 Highlight Main Ideas, 890–891 Identify Key Words and Context Clues, 290–291 Interpret Coordinate Graphs, 208–209 Interpret a Diagram, 510–511 Measure to Solve Problems, 736–737 Use a Formula Sheet, 646–647 Gridded Response: Record Your Answer, 136–137 Multiple Choice: Eliminate Answer Choices, 444–445 Recognize Distracters 816–817 Work Backward, 66–67 TAKS Tip, 67, 69, 137, 139, 209, 210, 291, 293, 373, 375, 445, 447, 511, 513, 579, 581, 647, 649, 737, 739, 817, 819, 891, 893 Tangency, point of, 746 Tangent, 525, 746, 805 to a circle at a point, constructing, 748 to |
Racing, Fort Worth, 392 Recreation, New Braunfels, 673 Space Shuttle, Houston, 548 Sports, Austin, 530 Transportation, Dallas, 183 Texas Motor Speedway, 392 Texas Star Ferris wheel, 841 Texas State Aquarium, 698 Texas State Capitol, 742 Texas state gemstone, 752 Textiles, 125 Theater, 246 Theorems, 110, 748–749, 757–758, 774–775, 782–784, 848–850 For a complete list, see pages S82–S87 proofs of, 753, 758, 762, 779, 783, 788–789 Theoretical probability, 628 Third Angles Theorem, 226 30°-60°-90° triangle, 358 drawing, 653 representations of, 661–664 surface area of, 680 Three dimensions Distance Formula in, 672 formulas in, 670–673 identifying symmetry in, 858 Midpoint Formula in, 672 using Pythagorean Theorem in, 671 Tick marks, 13 Tiling, 863, see also Tessellations Titan, 449 Tolerance, S72 Too much information, 209 Tools of Geometry, xxi Transformations, 50, 79 compositions of, see Compositions of transformations congruence, 824, 854 in the coordinate plane, 50–52 describing, in terms of reflections, 850 exploring, with geometry software, 56–57 of functions, 838 Math Builders, xxiv–xxv with matrices, 846–847 of parent functions, 838, S63 similarity, 873 using to create tessellations, 864 to extend tessellations, 870–871 Transit (tool), 20 Transitive Property, 168, 176 Transitive Property of Congruence, 106 Transitive Property of Equality, 104 Transitive Property of Inequality, 330 Transitive Property of Similarity, 473 Translation symmetry, 863 Translations, 50, 327, 831–833 constructing, 836 in the coordinate plane, 832 of figures, constructing, 831 general, in the coordinate plane, 832 horizontal, see Horizontal translations vertical, see Vertical translations Transportation, 183, 194, 360, 386, 620, 631, 633, 866 Transversals, 147 parallel lines and, angles formed by, 154–157 Trapezoid, 51, 426, 429, 819 area of, 590 base angles |
of, 429 bases of, 429 isosceles, see Isosceles trapezoids legs of, 429 midsegment of, 431 properties of, 429–431 proof of, 435 Tree rings, 604 Trend lines, S79 Trefoil shape, 313 Triangle(s), 36, 98, 216, 382 acute, 216 altitudes of, 314–317 defined, 316 angle bisectors of, 480 angle relationships in, 223–226 angle-side relationships in, 333 area of, 590 bisectors of, 307–310 centroid of, 314 constructing, 314 circumcenter of, 307 constructing, 307 circumscribe a circle about, 778 classifying, 216–219, 230 congruent, see Congruent triangles developing formulas for, 589–593 equiangular, 216 equilateral, see Equilateral triangles incenter of a, 309 isosceles, see Isosceles triangles medians of, 314–317 midsegment, 322 constructing, 327 musical, 218 obtuse, 216 orthocenter of a, 316 constructing, 320 right, see Right triangle(s) scalene, 217 solving, 535 special points in, 321 spherical, see Spherical triangles two, inequalities in, 340–342 Triangle Angle Bisector Theorem, 483 Triangle classification by angle measures, 216 by side lengths, 217 Triangle congruence, 212–295 applying ASA, AAS, and HL, 252–255 applying SSS and SAS, 242–245 CPCTC, 260–262 exploring SSS and SAS, 240–241 predicting other relationships, 250–251 Triangle inequalities, exploring, 331 Triangle Inequality Theorem, 334 proof of the, 338 Triangle Midsegment Theorem, 322–324 proof of the, 326 Triangle Proportionality Theorem, 481 Converse of the, 482 Triangle rigidity, 242 Triangle similarity AA, SSS, and SAS, 470–473 predicting, relationships, 468–469 Triangle Sum Theorem, 223 Trapezoid Midsegment Theorem, 431 Travel, 17, 54, 84, 335, 458, 484 Treadmill, 539 developing the, 222 proof of the, 223 Triangulation, 223 Index S175S175 Trigonometric functions, inverse, 533, Vertex, 20 534 Trigonometric ratios, 524–528 Trigonometry indirect measurement using, 550 right triangles and, 524–583 unit circle and, |
570–571 Trisecting angles, 25 Triskelion, 861 Trundle wheel, 605 Truth table, 128 Truth value, 82 Turns, see Rotations Two-column proofs, see Proofs, two-column Two-point perspective, 662 drawing figures in, 668 Two-Transversal Proportionality Corollary, 482 U Undefined terms, 6 Unit circle, 570 trigonometry and the, 570–571 University of Texas Longhorn Band, 833 Urban legends, 88 Use more than one method, 45 V Vanishing point, 662 Variation constant of, 501 direct, 501 Vasarely, Victor, 860 Vector(s), 559–563 addition of, 561–562 equal, 561 negation of a, 566 parallel, 561 resultant, 561 Venn diagrams, 80, S45, S81 of a cone, 690 of a polygon, 382 of a pyramid, 689 of a three-dimensional figure, 654 Vertex angles, 273 Vertical angles, 30 Vertical Angles Theorem, 120–122 proof of the, 120 Vertical line, equation of a, 190 Vertical translations in the coordinate plane, 832 of parent functions, 838 Vocabulary, 9, 17, 24, 31, 38, 47, 53, 77, 84, 91, 99, 107, 113, 122, 148, 185, 194, 219, 227, 234, 245, 256, 262, 270, 276, 304, 311, 317, 324, 336, 352, 386, 395, 412, 432, 457, 465, 491, 498, 521, 529, 547, 563, 603, 609, 633, 657, 665, 674, 684, 693, 701, 709, 718, 751, 760, 767, 776, 795, 827, 851, 859, 866, 875 math, learning, 299 Vocabulary Connections, 4, 72, 144, 214, 298, 378, 452, 516, 586, 652, 744, 822 Volume, 697, 789 of cones, 705–708 of cylinders, 697–700 of prisms, 697–700 of pyramids, 705–708 of spheres, 769 surface area and, comparing, 722–723 Vortex train, 867 |
W Washington, George, 18 What if...?, 26, 30, 108, 165, 183, 193, 230, 253, 275, 323, 349, 357, 359, 385, 424, 428, 456, 473, 495, 539, 545, 554, 556, 562, 673, 692, 698, 706, 793, 801, 825, 833, 849, 873 Whole numbers, 80 Working backward, 889, S43 Write About It Write About It questions appear in every exercise set. Some examples: 10, 18, 26, 33, 40 Write and solve an equation linear, 31–34, 38–41 literal, 41 Write Short Responses, 208–209 Writing equations, see Equations, writing Writing Math, 81, 96, 111, 463, 525, 756 Writing Strategies, see also Reading and Writing Math Draw Three-Dimensional Figures, 653 Write a Convincing Argument, 379 X x-axis, 42, S56 reflections across the, 826 x-coordinate, finding, 841 x-intercept, see Intercepts Y y-axis, 42, S56 reflections across the, 826 y-coordinate, finding, 841 y-intercept, see Intercepts Z ZDecimal, 189 ZSquare, 189 ZStandard, 189 Zilker Kite Festival, 428 Zulu people, 834 S176 S176 Index Credits Credits Abbreviations used: (t) top, (c) center, (b) bottom, (l) left, (r) right, (bkgd) background rights reserved.; 193 (tr), Alamy Images; 195 Sipa Photos/Newscom.com; 196 (tl), Comstock/Fotosearch; 200 (tl), Comstock/Fotosearch; 200 (tr), Pierre Vivant Staff Credit Bruce Albrecht, Angela Beckmann, Nancy Behrens, Lorraine Cooper, Marc Cooper, Lana Cox, Jennifer Craycraft, Martize Cross, Nina Degollado, Lydia Doty, Sam Dudgeon, Kelli R. Flanagan, Mary Fraser, Stephanie Friedman, Jeff Galvez, Pam Garner, Diannia Green, Tracie Harris, Tessa Henry, Liz Huckestein, Jevara Jackson, Kadonna Knape, Cathy Kuhles, Jill M. Lawson, Peter Leighton, Christine MacInnis, Jonathan Mart |
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Royalty Free; 19 (c), Gabriel Bouys/AFP/Getty Images; 20 (tr), Gary Conner/Photo Edit; 21 (tl), HRW Photo; 26 (tl), Photodisc Royalty Free; 28 (t), Jon Feingersh/CORBIS; 32 (bl), Photodisc Royalty Free; 34 (tl), Photodisc Royalty Free; 34 (br), Gibson Stock Photography; 38 (c-triangle), Victoria Smith/HRW Photo; 39 (bl), HRW Photo; 40 (l), Alamy Images; 41 (l), The Bridgeman Art Library/ Getty Images; 43 (tl), HRW Photo; 43 (tr), Russell Andorka; 46 (tl), Duomo/CORBIS; 48 (cl), Sam Dudgeon/HRW; 48 (bl), HRW Photo by Sam Dudgeon; 54 (tl), BrandX/ Fotosearch.com; 54 (bl), HRW Photo; 58 (tl), HRW Photo by Sam Dudgeon; 58 (c), Marty Granger/HRW Photo Chapter 2: 70–71 Courtesy of 7A Ranch; 74 (tr), Stephen Frink/CORBIS; 76 NASA Images; 78 (tl), Index Stock/Picturequest; 78 (bl), Victoria Smith/HRW Photo; 78 (br), The Granger Collection; 81 (tr), BARRY RUNK/STAN/Grant Heilman Photo; 83 (cl), Digital Vision/Getty; 83 (cr), Picturequest Images; 85 (bl), Victoria Smith/HRW Photo; 85 (br), Corbis; 86 Mark Sykes/Alamy Images; 87 (bl), Michael Newman/PhotoEdit; 88 (tr), Alamy Images; 88 (br), Taxi/Getty Images; 92 (cl), Daniel Miller/AP Wide World; 92 (bl), Victoria Smith/HRW Photo; 95 (br), Sam Dudgeon/HRW Photo; 96 Darrell Gulin/CORBIS; 99 Getty/Stone; 100 (tl), Nibsc/Photo Researchers, Inc.; 100 (bl) Victoria Smith/HRW; 100 (br), The Granger Collection; 102 (tl), HRW/Victoria Smith; 102 (c-Alice and Chesire Cat), Corbis Images; 102 (br), (tr), The Pierpont Morgan Library/Art Resource, NY; 104 Getty Images; 109 (tl), Photo-objects/Fotose |
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), Corbis Images/Punchstock.com; 360 (br), Sam Dudgeon/HRW Photo; 361 (tl), PhotoStockFile/Alamy; 361 (cr), HRW Photo; 361 (bl), Transtock Inc./Alamy Images; 364 (tl), Transtock Inc./Alamy Images; 364 (b), Paul Doyle/Alamy Chapter 6: 376–377 Donne Bryant/Art Resource; 382 (tr), AP Photo/Pat Sullivan; 385 (cr), Custom Medical Stock; 386 (tl), Ingram Image/Picturequest; 386 (tcl), Punchstock; 386 (tr), Comstock/Punchstock; 386 (tcr), Alamy Images; 386 (cr), Alamy Images; 387 (tr), Peter Van Steen/HRW Photo; 387 (bl), Sam Dudgeon/HRW Photo; 387 (br), HRW Photo by Peter Van Steen; 390 (t, c, b), HRW Photo; 391 (tr), George D. Lepp/CORBIS; 392 (cl), Michael Kim/Corbis; 392 (br), Art Reference: BasketballHoops Unlimited; 395 (tr), Robert Harding World Imagery/Getty Images; 395 (cr), Photo Edit Inc.; 396 (tr), Photo Courtesy Mohawk Lifts; 396 (bl), HRW Photo; 396 (br), HRW Photo by Peter Van Steen; 398 (tr), Sam Dudgeon/HRW; 401 (cl), Cottonwood Photography; 401 (cr), Sam Dudgeon/HRW; 402 (cr-parallel rule) Victoria Smith/HRW; 402 (cr-map), Alamy Images; 403 (tr)(inset), Sam Dudgeon/HRW; 404 (cl), Historical Picture Archive/CORBIS; 404 (cr), Robervals’s balance/Varnished wood, brass French, Late 18th century/700 mm x 360 mm; diam. 160 mm Inventory number 1983.26.36, Stewart Museum, Montreal, Canada; 404 (bl), Sam Dudgeon/HRW Photo; 404 (br), HRW Photo by Peter Van Steen; 405 (cr), Stinar Corporation; 406 (bl), Guinea REUTERS/Corbis Images; 406 (tl), HRW Photo; 407 (tr-Flourite) (cl-Rhodochrosite) (inset), Chip Taylor/NMNH/Smithsonian Institution; 407 (cr) |
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AP/Wide World Photos; 463 (bl), ©Dennis Boissavy/Getty Images; 464 (t), ©Nathan Keay/HRW; 465 (cr), ©Cameron Cross; 466 (tr), ©Nathan Keay/HRW Photo; 466 (cl), OwakiKulla/CORBIS; 466 (bl), ©Hemera Technologies/Alamy Photos; 470 (tr), RoyaltyFree/CORBIS; 472 (b), PhotoDisc/Getty Images; 476 (cl), AFP PHOTO/NOAA/NewsCom; 476 (tl), ©Hemera Technologies/Alamy Photos; 478 (tl), ©Hemera Technologies/ Alamy Photos; 478 (cr), ©David Young-Wolff/PhotoEdit; 481 (tr), ©Christie’s Images/ CORBIS; 486 (tl), Royalty-Free/Comstock; 488 (tr), roadsideamerica.com, Kirby, Smith & Wilkins; 489 (br), ©Peter Gridley/Getty Images; 491 (br), Courtesy NASA/JPL/ASU; 492 (cl), ©Reuters/CORBIS; 492 (bl), Royalty-Free/Comstock; 493 (tl), ©Sygma/ CORBIS; 493 (tr), ©Tom Bean/CORBIS; 494 (tr), ©Jack Van Dusen; 494 (bl), ©Mark Richards/PhotoEdit; 495 (tr), David Neuse Photography; 495 (cr), ©Photodisc/Getty Images; 495 (br), Darrell Gulin/CORBIS; 499 (bl), Royalty-Free/Comstock; 502 (tr), Comstock/HRW Photo Research Library; 502 (cr), ©Jeff Cadge/Getty Images Chapter 8: 514–515 (bkgd), Painet Inc.; 518 (tr), Twisted Texas/Wesley Treat; 522 (bl), Fotosearch; 525 (tr), Alamy Images; 529 (br), Alamy Images; 530 (tl), Phil Gustafson; 530 (cr), Kevin Fleming/Corbis; 530 (bl), Fotosearch; 531 (tl), Alamy Images; 534 (tr), Photo Edit Inc.; 535 (bl), PhotoEdit; 536 (c), Getty Images; 5 |
38 (tr), Getty Images Sport/Bobby Julich; 539 (tl), Fotosearch; 539 (cl), Photo Edit Inc.; 542 (tl), Fotosearch; 542 (b), Superstock; 544 (tr), Stone/Getty Images; 548 (tl), Scott Berner/Index Stock Imagery, Inc.; 548 (bl), Fotosearch; 550 (tr), HRW Photo; 550 (cr), HRW Photo; 551 (tr), Alamy Images; 556 (tl), Brad Smith/News & Observer/AP/Wide World Photos; 557 (tl), Fotosearch; 559 (tr), Stone/Getty; 565 (tl), Fotosearch; 566 (t), FOXTROT ©1999 Bill Amend. Reprinted with permission of UNIVERSAL PRESS SYNDICATE. All rights reserved.; 566 (cl), Photodisc/Picturequest; 568 (tr), Fotosearch; 568 (b), Stone/Getty Images; 582 (b), Chad Ehlers/PictureQuest; 582 (tr), Jim Olive/ Stockyard.com; 583 (tc), Yoichi R. Okamoto/LBJ Library Photo; 583 (bl), Steve Warble/ Mountain Magic Photography; 583 (cr), National Park Service Chapter 9: 584 Jim Wark; 589 (tr), Victoria Smith/HRW Photo; 594 (c), HRW Photo by Sam Dudgeon; 594 (cr), HRW Photo by Sam Dudgeon; 595 (cl), The Granger Collection, New York; 595 (tl), Photodisc/Getty Images; 596 (cr), HRW Photo; 598 (all) Sam Dudgeon/HRW Photo; 600 (tr), ©gkphotography/Alamy Photos; 604 (cl), ©Royalty-Free/CORBIS; 604 (bl), ©Photodisc/Getty Images; 606 (tr), ©Rose/Zefa/ Masterfile; 607 (bl), Courtesy of Texas Highways Magazine; 610 (bl), ©Photodisc/ Getty Images; 612 (bl), ©Royalty Free/CORBIS; 614 (tr), Thinkstock/PictureQuest; 614 (c), ©Otto Rogge/CORBIS; 620 (bl), ©Royalty-Free/CORBIS |
; 626 (cl), ©Patrick Ray Dunn/Alamy Photos; 626 (bl), ©Royalty-Free/CORBIS; 628 (c), Peter Van Steen/ HRW; 629 (t), Peter Van Steen/HRW; 630 (tr), AP Photo/Reed Saxon; 628 (cr), Peter Van Steen/HRW; 632 (tl), Warren Morgan/CORBIS; 635 (tl), Romeo Gacad/AFP/Getty Images; 635 (bl), Royalty-Free/CORBIS; 637 (cr), HRW Photo by Sam Dudgeon; 638 (tl), Royalty-Free/CORBIS; 638 (b), Photofusion Picture Library/Alamy Photos; 638 (cr), Dennis MacDonald/PhotoEdit Chapter 10: 650–651 Doug Hopfer/HRW; 654 (tr), AFP PHOTO/JIJI PRESS/Newscom; 655 (tl), Fotosearch; 655 (tr), David Young-Wolff/Photo Edit; 655 (cl), HRW Photo; 655 (cr), HRW Photo; 656 (cr), Newscom; 657 (tl) Bonillo/Photo Edit; 657 (tc), Photodisc/RF/Fotosearch; 657 (tr), HRW Photo; 658 (bl), David Young-Wolff/Photo Edit; 661 (cr, br), HRW Photo; 662 (tr, cr, bc), HRW Photo; 662 (bl), Corbis Images; 664 (tr, cr), Victoria Smith/HRW Photo; 665 (tl) Victoria Smith/HRW Photo; 665 (all cubes), Victoria Smith/HRW Photo; 666 (tl, tc, tr, cr), Victoria Smith/HRW Photo; 666 (bl), David Young-Wolff/Photo Edit; 667 (tr)–poster, Duomo/CORBIS; 670 (tr), Jeff Hunter/The Image Bank/Getty Images; 673 (cl), Peter Essick/Getty Images; 675 (bl), David Young-Wolff/Photo Edit; 675 (tl), Stone/Getty Images; 678 (tl), David Young Wolf/Photoedit; 678 (b), Courtesy of Texas Highways Magazine; 679 (cr), Victoria Smith/HRW Photo; 680 (tr), |
Robert Harding World Imagery/Getty Images; 683 (cl), Creative Ice Carvings; 686 (bl) HRW Photo; 687 (bl)(bc)(br), HRW Photo; 688 (all), HRW Photo; 692 (cl), Marc Golub/HRW Photo; 695 (tl), Victoria Smith/HRW; 695 (cl), Dennis MacDonald/Photo Edit; 697 (tr), Jeff Greenberg/Photoedit Inc.; 697 (tc), HRW Photo; 698 (cl), ©John Elk III; 699 (tc)(tr), HRW Photo; 701 (cr), AFP/TIMOTHY A. CLARY/Getty Images; 703 (tl), Victoria Smith/HRW Photo; 703 (cl), AKG Images; 703 (cr), HRW Photo; 705 (tr), ©Mark Gibson Photography; 706 (cr), ©Lyndol Descant/LyndolDotCom; 711 (bl), HRW Photo; 718 (cl)(cr), Victoria Smith/HRW Photo; 719 (tr), ©Susan Van Etten/Photo Edit; 720 (Chart-Golf Ball), Martin Paul Ltd., Inc./ Picturequest; 720 (Chart-Cricket ball), Photodisc/RF/Fotosearch; 720 (Chart-Tennis ball), Stockdisc/RF/Getty Images; 720 (Chart-Petanque ball), Dk Images/RF/Getty Images; 720 (cl), Ralph White/CORBIS; 720 (bl), HRW Photo; 721 (cr), Lyndol Descant/ LyndolDotCom; 724 (tl), HRW Photo; 724 (b), Picturequest; 726 (bl), HRW Photo; 729 (tcl) (tcr), ©Dorling Kindersley/Getty Images; 729 (cr), Royalty-Free/Alamy Images; 729 (br), Stockbyte/RF/Picturequest; 731 (tl, tr, c), Victoria Smith/HRW Photo; 734 (tc), HRW Photo; 740 (tr), ©Space Imaging; 740 (bl), Alberto Tamargo/Getty Images; 740 (br), ©Cut and Deal Ltd/Alamy; 741 (tc), United States Mint image; 741 (cr), Lyle Engleson/Goldberg Coins; 741 (cl, |
bc), Jerry Adams Chapter 11: 742–743 Victoria Smith/HRW; 746 (tr), ©NASA/Roger Ressmeyer/ CORBIS; 749 (tr), ©Alan Kearney/Getty Images; 749 (br), Gamma; 752 (cl), ©CORBIS; 752 (bc), Courtesy of Texas Highways Magazine; 753 (bl), Photolibrary.com.pty. ltd./Index Stock Imagery, Inc.; 756 (tr), ©Brand X Pictures/PunchStock; 762 (bl), ©photolibrary.com.pty.ltd./Index Stock Imagery, Inc.; 762 (c), Victoria Smith/HRW Photo; 764 (tr), AP/Wide World Photos; 764 (tr), Jim Wark/Airphoto; 767 (br), ©Christer Fredriksson/Lonely Planet Images; 768 (tl) ©Photolibrary.com.pty.ltd./Index Stock Imagery, Inc.; 768 (b), ©Tony Freeman/PhotoEdit; 768 (cl), Scala/Art Resource, NY; 768 (bl), ©Photolibrary.com.pty.ltd./Index Stock Imagery, Inc.; 768 (br), Photo by Eisenmann, N.Y./Library of Congress; 772 (tr), Victoria Smith/HRW Photo; 773 (br), Victoria Smith/HRW Photo; 776 (tc), Victoria Smith/HRW Photo; 777 (bl), Victoria Smith/HRW Photo; 777 (cr), H. Armstrong Roberts/RobertStock.com; 778 (tr), ©Archivo Iconografico, S.A./CORBIS; 787 (b) Linda Owen; 788 (bl) Victoria Smith/ HRW Photo; 792 (tr), ©Joathan Blair/CORBIS; 795 (br), ©Chris Lisle/CORBIS; 796 (cr), ©Michael T. Sedam/CORBIS; 797 (tl), NASA Marshall Space Flight Center (NASAMSFC); 797 (bl), Victoria Smith/HRW Photo; 799 (tr), Cartoon copyrighted by Mark Parisi, printed with permission.; 800 (bl), ©RuberBall/Alamy Photos; 802 (br) ©Alamy Photos; 803 (tl), www.lonestarthrills.com; 803 (bl), Victoria Smith |
/HRW Photo; 804 (tl), The Granger Collection, New York; 805 (bl), Sam Dudgeon/HRW; 806 (tr), Victoria Smith/HRW Photo Chapter 12: 820–821 ©Royalty-Free/Corbis; 823 (tl), Sam Dudgeon/HRW; 823 (tcl), Sam Dudgeon/HRW; 823 (bcl), Sam Dudgeon/HRW; 823 (bl), Sam Dudgeon/HRW; 824 (tr), Scott Teven/photohouston; 824 (cr), Sam Dudgeon/HRW; 824 (c), Sam Dudgeon/HRW; 824 (cl), Sam Dudgeon/HRW; 824 (br), Sam Dudgeon/HRW; 828 (cr), Musee d’Orsay, Paris, France/Erich Lessing/Art Resource, NY; 829 (tl), ©Brian Hagiwara/Brand X Pictures/Getty Images; 831 (tr), ©Steve Boyle/NewSport/CORBIS; 831 (cl), Sam Dudgeon/HRW; 831 (c), Sam Dudgeon/HRW; 831 (cr), Sam Dudgeon/ HRW; 831 (br), Sam Dudgeon/HRW; 833 (cl), ©Kelly-Mooney Photography/CORBIS; 834 (cr), Bonhams, London, UK/The Bridgeman Art Library; 835 (tl), Photo by Walt Disney Studios/ZUMA Press ©Copyright 1998 by Courtesy of Walt Disney Studios; 835 (bl), ©Brian Hagiwara/Brand X Pictures/Getty Images; 836 (tr), Victoria Smith/ HRW; 839 (tr), ©Roger Ressmeyer/CORBIS; 839 (cl, c, cr, br), Sam Dudgeon/HRW; 841 (cl), ©L. Clarke/CORBIS; 843 (bl), ©Brian Hagiwara/Brand X Pictures/Getty Images; 844 (tr), ©Roger Ressmeyer/CORBIS; 848 (tr), Scott Teven/photohouston; 853 (tl), ©Brian Hagiwara/Brand X Pictures/Getty Images; 854 (tl), ©Brian Hagiwara/ Brand X Pictures/Getty Images; 854 (br), ©Brand X Pictures/Getty Images; 856 |
(tr), Jan Hinsch/Photo Researchers, Inc.; 856 (br), ©One Mile Up, Inc; 857 (c-purple diatoms), Alfred Pasieka/Photo Researchers, Inc.; 857 (bc), Eric Grave/Photo Researchers, Inc.; 857 (br), John Burbidge/Photo Researchers, Inc.; 859 (cr), spaceimaging.com/Getty Images; 859 (br), 859 (br), ©Brand X Pictures/Alamy Photos; 859 (tr), M. C. Escher’s “Circle Limit III” ©2005 The M.C. Escher CompanyHolland. All rights reserved. www.mcescher.com; 860 (tr), (c) ARS, NY/Art Resource, NY; 861 (tl), M. C. Escher’s Wooden Ball with Fish ©2005 The M.C. Escher CompanyHolland. All rights reserved. www.mcescher.com; 861 (cr), Comstock Images/ PictureQuest; 861 (cr), ©bildagentur-online.com/de/Alamy Photos; 863 (tr), ©Russell Gordon/DanitaDelimont.com; 863 (tcl), ©Anna Zuckerman-Vdovenko/PhotoEdit; 863 (tcr), ©Danita Delimont/Alamy Photos; 863 (cl) ©Paul Souders/WorldFoto/ Alamy Photos; 863 (cr), ©Jeff Greenberg/Alamy Photos; 863 (cbl)(cbr), ©Danita Delimont/Alamy Photos; 865 (tl), ©Comstock Images/Alamy Photos; 865 (cl), Darren Matthews/Photographer’s Direct; 865 (c), ©G. Schuster/Photo-AG/CORBIS; 865 (cr), Brand X Pictures/PictureQuest; 866 (bc), ©M. Angelo/CORBIS; 866 (br), ©Paul Almasy/CORBIS; 866 (bl), Wolfgang Kaehler Photography; 867 (all), Sam Dudgeon/HRW; 868 (tl), M. C. Escher’s Wooden Ball with Fish ©2005 The M.C. Escher Company-Holland. All rights reserved. www.mcescher.com |
; 868 (tr); M. C. Escher’s “Symmetry Drawing E103” ©2005 The M.C. Escher Company-Holland. All rights reserved. www.mcescher.com; 868 (tcl), M. C. Escher’s “Verbum” ©2005 The M.C. Escher Company-Holland. All rights reserved. www.mcescher.com; 868 (tcr), M. C. Escher’s “Symmetry Design E38” ©2005 The M.C. Escher Company-Holland. All rights reserved. www.mcescher.com; 868 (cl, br), Sam Dudgeon/HRW; 870 (tr, cr, br), Sam Dudgeon/HRW; 871 (all), Sam Dudgeon/HRW; 872 (tr), Mark Lennihan/AP/Wide World; 876 (tl), Texas State Library & Archives Commission; 876 (bl), M. C. Escher’s Wooden Ball with Fish ©2005 The M.C. Escher Company-Holland. All rights reserved. www.mcescher.com; 876 (br), M. C. Escher’s “Drawing Hands” ©2005 The M.C. Escher Company-Holland. All rights reserved. www.mcescher.com; 876 (tl), AP Photo/ The Truth, Jennifer Shephard; 877 (cl)(cr), Adam Hart-Davis/Photo Researchers, Inc.; 880 (tcl), M. C. Escher’s “Symmetry Drawing E93” ©2005 The M.C. Escher CompanyHolland. All rights reserved. www.mcescher.com; 880 (c), ©Alamy Photos; 881 (tl), M. C. Escher’s Wooden Ball with Fish ©2005 The M.C. Escher Company-Holland. All rights reserved. www.mcescher.com; 880 (tcr), M. C. Escher’s “Symmetry Drawing E91” ©2005 The M.C. Escher Company-Holland. All rights reserved. www.mcescher. com; 880 (bcl), |
M. C. Escher’s “Path of Life III” ©2005 The M.C. Escher CompanyHolland. All rights reserved. www.mcescher.com; 880 (bcr), M. C. Escher’s “Symmetry Drawing E69” ©2005 The M.C. Escher Company-Holland. All rights reserved. www. mcescher.com; 880 (b), M. C. Escher’s “Reptiles” ©2005 The M.C. Escher CompanyHolland. All rights reserved. www.mcescher.com; 882 (tr), GEORGE POST/Photo Researchers, Inc.; 883 (cr), ©George W. Hart; 894 (cr), ©Buddy Mays/CORBIS; 894 (bl), Courtesy of the Smithsonian Institution, NMAH/Transportation; 895 (bc), Stephanie Friedman/HRW; 895 (cr), Courtesy of Texas Department of Transportation Back Matter: S2 Don Couch/HRW; S3 John Langford/HRW S178 S178 Credits a207se_toc_vii-xix.sw.indd viii a207se_toc_vii-xix.sw.indd viii 11/29/05 4:47:45 PM 11/29/05 4:47:45 PM two outputs, 1 and and 9, also have more than one output, 3 26 1. is not a function Two other inputs, 4 1, 1 2 1,, 0, 0 1, 1, 1 51 Although 1 appears as an output twice, each input has one and only one output. is a function. 4, 2 9, 3 26,,,, 2 9, 3, 0, 0 4, 2 51 because each input corresponds to one and only one output. 9, 3 1, 1 9, 3 1, 1 4, 2 26,,,,, is a function ■ Calculator Exploration Make a scatter plot of each set of ordered pairs in Example 4. Examine each scatter plot to determine if there is a graphical test that can be used to determine if each input produces one and only one output, that is, if the set represents a function. Example 5 Finding Function Values from a Graph The graph in Figure 1.1-8 defines a function whose rule is: For input x, |
the output is the unique number y such that (x, y) is on the graph. y 4 2 −2 −4 −2 x 2 4 Figure 1.1-8 a. Find the output for the input 4. b. Find the inputs whose output is 0. x 2. c. Find the y-value that corresponds to d. State the domain and range of the function. Solution a. From the graph, if x 4, then y 3. Therefore, 3 is the output corresponding to the input 4. Section 1.1 Real Numbers, Relations, and Functions b. When y 0, x 3 or x 0 or x 2. Therefore, 3, 0, and 2 are the inputs corresponding to the output 0. x 2 c. The y-value that corresponds to d. The domain is all real numbers between is 4 y 3. range is all real numbers between and 3, inclusive. 2 and 5, inclusive. The 9 ■ CAUTION The parentheses in d(t) do not denote multiplication. The entire symbol d(t) is part of the shorthand language that is convenient for representing a function, its input and its output; it is not the same as algebraic notation. Function Notation Because functions are used throughout mathematics, function notation is a convenient shorthand developed to make their use and analysis easier. Function notation is easily adapted to mathematical settings, in which the particulars of a relationship are not mentioned. Suppose a function is given. Let f denote a given function and let a denote a number in the domain of f. Then f(a) denotes the output of the function f produced by input a. For example, f(6) denotes the output of the function f that corresponds to the input 6. y is the output produced by input x according to the rule of the function f NOTE The choice of letters that represent the function and input may vary. is abbreviated which is read “y equals f of x.’’ y f x, 2 1 In actual practice, functions are seldom presented in the style of domainrule-range, as they have been here. Usually, a phrase, such as “the function of directions, as shown in the following diagram. 2x2 1, x f 2 1 ” will be given. It should be understood as a set Name of function Input number > > 2x2 1 f x ⎧⎨⎩ ⎧⎪⎨⎪⎩ |
2 > 1 > Output number Directions that tell you what to do with input x in order to produce the corresponding output f(x), namely, “square it, add 1, and take the square root of the result.” For example, to find f(3), the output of the function f for input 3, simply replace x by 3 in the rule’s directions. 2x2 Similarly, replacing x by 29 1 210 1 5 2 and 0 produces the respective outputs. 202 1 1 2 1 226 and f 0 1 2 10 Chapter 1 Number Patterns Technology Tip is to enter 2 1 y f One way to evaluate a x function f its rule into the equation memory as and use TABLE or EVAL. See the Technology Appendix for more detailed information. x 1 2 NOTE Functions will be discussed in detail in Chapter 3. Example 6 Function Notation For h a. h A x 2 1 23 x2 x 2, find each of the following: B b. h 1 2 2 c. h 1 a 2 Solution h To find rule of h. A 23 B and h 2 1 2, replace x by 23 and 2, respectively, in the a. b. h h 23 A 2 1 2 B 1 23 A 2 B 2 2 2 23 2 3 23 2 1 23 2 1 2 2 4 2 2 0 The values of the function h at any quantity, such as can be found by using the same procedure: replace x in the formula for h(x) by the quantity a a, and simplify. a 2 1 2 a2 a 2 ■ In Exercises 6–8, sketch a scatter plot of the given data. In each case, let the x-axis run from 0 to 10. 6. The maximum yearly contribution to an individual retirement account in 2003 is $3000. The table shows the maximum contribution in fixed 2003 dollars. Let correspond to 2000. x 0 Year 2003 2004 2005 2006 2007 2008 Amount 3000 2910 3764 3651 3541 4294 7. The table shows projected sales, in thousands, of x 0 personal digital video recorders. Let correspond to 2000. (Source: eBrain Market Research) Year 2000 2001 2002 2003 2004 2005 Sales 257 129 143 214 315 485 Exercises 1.1 1. Find the coordinates of points A–I In Exercises 2–5, find the coordinates of the point P. 2. P lies 4 units to the left of the y-axis and 5 units below the x-axis. |
3. P lies 3 units above the x-axis and on the same. vertical line as 6, 7 1 2 4. P lies 2 units below the x-axis and its x-coordinate is three times its y-coordinate. 5. P lies 4 units to the right of the y-axis and its y-coordinate is half its x-coordinate. 8. The tuition and fees at public four-year colleges in the fall of each year are shown in the table. Let x 0 correspond to 1995. (Source: The College Board) Section 1.1 Real Numbers, Relations, and Functions 11 Tuition Year & fees Tuition Year & fees 1995 $2860 1998 $3247 1996 $2966 1999 $3356 1997 $3111 2000 $3510 9. The graph, which is based on data from the U.S. Department of Energy, shows approximate average gasoline prices (in cents per gallon) between 1985 and 1996, with corresponding to 1985. x 0 y 120 100 80 60 40 20 1 2 3 4 5 6 7 8 9 10 11 x a. Estimate the average price in 1987 and in 1995. b. What was the approximate percentage increase in the average price from 1987 to 1995? c. In what year(s) was the average price at least $1.10 per gallon? 10. The graph, which is based on data from the U.S. Department of Commerce, shows the approximate amount of personal savings as a percent of disposable income between 1960 and 1995, with x 0 corresponding to 1960. y 10 10 15 20 25 30 35 x a. In what years during this period were personal savings largest and smallest (as a percent of disposable income)? b. In what years were personal savings at least 7% of disposable income? 11. a. If the first coordinate of a point is greater than 3 and its second coordinate is negative, in what quadrant does it lie? b. What is the answer to part a if the first coordinate is less than 3? 12. In which possible quadrants does a point lie if the product of its coordinates is b. negative? a. positive? 13. a. Plot the points (3, 2), 4, 1 1, 2 1 2, 3, and 2 5, 4. 2 1 b. Change the sign of the y-coordinate in each of the points in part a, and plot these new points. c. Explain how the points (a, b) and a |
, b are 2 1 related graphically. Hint: What are their relative positions with respect to the x-axis? 14. a. Plot the points (5, 3), 4, 2 1, 2 1 1, 4, and 2 3, 5. 2 1 b. Change the sign of the x-coordinate in each of the points in part a, and plot these new points. c. Explain how the points (a, b) and a, b are 2 1 related graphically. Hint: What are their relative positions with respect to the y-axis? In Exercises 15 – 18, determine whether or not the given table could possibly be a table of values of a function. Give reasons for each answer. 15. 16. 17. 18. Input Output Input Output Input Output Input Output.5 0 0 3 4 2 ± 14 5 3 7 8 3 8 12 Chapter 1 Number Patterns Exercises 19 – 22 refer to the following state income tax table. Annual income Amount of tax Less than $2000 0 $2000–$6000 2% of income over $2000 More than $6000 $80 plus 5% of income over $6000 19. Find the output (tax amount) that is produced by each of the following inputs (incomes): $500 $6783 $1509 $12,500 $3754 $55,342 20. Find four different numbers in the domain of this function that produce the same output (number in the range). 21. Explain why your answer in Exercise 20 does not contradict the definition of a function. 22. Is it possible to do Exercise 20 if all four numbers in the domain are required to be greater than 2000? Why or why not? 23. The amount of postage required to mail a firstclass letter is determined by its weight. In this situation, is weight a function of postage? Or vice versa? Or both? 24. Could the following statement ever be the rule of a function? For input x, the output is the number whose square is x. Why or why not? If there is a function with this rule, what is its domain and range? Use the figure at the top of page 13 for Exercises 25–31. Each of the graphs in the figure defines a function. 25. State the domain and range of the function defined by graph a. 26. State the output (number in the range) that the function of Exercise 25 produces from the following inputs (numbers in the domain): 2, 1 |
, 0, 1. 27. Do Exercise 26 for these numbers in the domain: 1 2, 5 2, 5 2. 28. State the domain and range of the function defined by graph b. 29. State the output (number in the range) that the function of Exercise 28 produces from the following inputs (numbers in the domain): 2, 0, 1, 2.5, 1.5. 30. State the domain and range of the function defined by graph c. 31. State the output (number in the range) that the function of Exercise 30 produces from the following inputs (numbers in the domain): 2, 1, 0,, 1. 1 2 32. Find the indicated values of the function by hand and by using the table feature of a calculator. 2x 4 2 x g 2 1 2 a. g d. g(5) 1 2 b. g(0) e. g(12) c. g(4) 33. The rule of the function f is given by the graph. Find a. the domain of f b. the range of f c. d. e. f4 −3 −2 −1 1 2 3 4 −2 −3 −4 34. The rule of the function g is given by the graph. Find a. the domain of g b. the range of g c. g d. g e. g f4 −3 −2 −1 1 2 3 4 −2 −3 − Section 1.2 Mathematical Patterns 13 y y y 4 3 2 1 −1 −1 −2 −3 −4 −3 −2 x 1 2 3 −3 −2 4 3 2 1 −1 −1 −2 −3 −4 x 1 2 3 4 −3 −2 4 3 2 1 −1 −1 −2 −3 −4 x 1 2 3 a. b. c. 1.2 Mathematical Patterns Objectives • Define key terms: sequence sequence notation recursive functions • Create a graph of a sequence • Apply sequences to real- world situations Definition of a Sequence Visual patterns exist all around us, and many inventions and discoveries began as ideas sparked by noticing patterns. Consider the following lists of numbers. 4, 1, 2, 5, 8,? 1, 10, 3, 73,? Analyzing the lists above, many people would say that the next number in the list on the left is 11 because the pattern appears to be “add 3 to the previous term.� |
�’ In the list on the right, the next number is uncertain because there is no obvious pattern. Sequences may help in the visualization and understanding of patterns. A sequence is an ordered list of numbers. Each number in the list is called a term of the sequence. An infinite sequence is a sequence with an infinite number of terms. Examples of infinite sequences are shown below. 2, 4, 6, 8, 10, 12, p 1, 3, 5, 7, 9, 11, 13, p 2 5, p 2, 1, 2 3 2 7 2 6 2 4,,,, The three dots, or points of ellipsis, at the end of a sequence indicate that the same pattern continues for an infinite number of terms. A special notation is used to represent a sequence. 14 Chapter 1 Number Patterns Sequence Notation The following notation denotes specific terms of a sequence: • The first term of a sequence is denoted u2. • The second term • The term in the nth position, called the nth term, is u1. denoted by un. • The term before un is un1. NOTE Any letter can be used to represent the terms of a sequence. Example 1 Terms of a Sequence Make observations about the pattern suggested by the diagrams below. Continue the pattern by drawing the next two diagrams, and write a sequence that represents the number of circles in each diagram. Diagram 1 1 circle Diagram 2 3 circles Diagram 3 5 circles Solution Adding two additional circles to the previous diagram forms each new diagram. If the pattern continues, then the number of circles in Diagram 4 will be two more than the number of circles in Diagram 3, and the number of circles in Diagram 5 will be two more than the number in Diagram 4. Diagram 4 7 circles Diagram 5 9 circles The number of circles in the diagrams is represented by the sequence 1, 3, 5, 7, 9, p, 5 6 Technology Tip If needed, review how to create a scatter plot in the Technology Appendix. which can be expressed using sequence notation. u1 1 u2 3 u3 5 u4 7 u5 9 p un un1 2 ■ Graphs of Sequences A sequence is a function, because each input corresponds to exactly one output. • The domain of a sequence is a subset of the integers. • The range is the set of terms of the sequence. Section 1.2 Mathematical Patterns 15 Because the domain of a sequence is discrete, the |
graph of a sequence consists of points and is a scatter plot. Example 2 Graph of a Sequence Graph the first five terms of the sequence 1, 3, 5, 7, 9, p 5. 6 Solution The sequence can be written as a set of ordered pairs where the first coordinate is the position of the term in the sequence and the second coordinate is the term. 10 (1, 1) (2, 3) (3, 5) (4, 7) (5, 9) Figure 1.2-1 The graph of the sequence is shown in Figure 1.2-1. ■ Recursive Form of a Sequence In addition to being represented by a listing or a graph, a sequence can be denoted in recursive form. 10 0 0 Recursively Defined Sequence A sequence is defined recursively if the first term is given and there is a method of determining the nth term by using the terms that precede it. Example 3 Recursively Defined Sequence Define the sequence 7, 4, 1, 2, 5, p 5 6 recursively and graph it. Solution 10 0 10 10 Figure 1.2-2 The sequence can be expressed as 7 u2 4 u3 u1 1 u4 2 u5 5 The first term is given. The second term is obtained by adding 3 to the first term, and the third term is obtained by adding 3 to the second term. Therefore, the recursive form of the sequence is 7 n 2. 3 and for un1 un u1 The ordered pairs that denote the sequence are 1, 7 2, 4 3, 1 2 1 The graph is shown in Figure 1.2-2. 1 1 2 2 4, 2 1 2 5, 5 1 2 ■ 16 Chapter 1 Number Patterns Calculator Exploration An alternative way to think about the sequence in Example 3 is Each answer Preceding answer 3. • Type 7 into your calculator and press ENTER. This establishes the first answer. • To calculate the second answer, press to automatically place ANS at the beginning of the next line of the display. • Now press 3 and ENTER to display the second answer. • Pressing ENTER repeatedly will display subsequent answers. Figure 1.2-3 See Figure 1.2-3. Technology Tip Alternate Sequence Notation Sometimes it is more convenient to begin numbering the terms of a sequence with a number other than 1, such as 0 or 4. The sequence graphing mode can be found in the TI MODE menu or the RECUR submenu |
of the Casio main menu. On such calculators, recursively defined function may be entered into the sequence memory, or your instruction manual for the correct syntax and use. Y list. Check u0, u1, u2, p or b4, b5, b6, p Example 4 Using Alternate Sequence Notation A ball is dropped from a height of 9 feet. It hits the ground and bounces to a height of 6 feet. It continues to bounce, and on each rebound it rises to 2 3 the height of the previous bounce. a. Write a recursive formula for the sequence that represents the height of the ball on each bounce. b. Create a table and a graph showing the height of the ball on each bounce. c. Find the height of the ball on the fourth bounce. Solution a. The initial height, u0, is 9 feet. On the first bounce, the rebound height, u1, is 6 feet, which is 2 3 the initial height of 9 feet. The recursive form of the sequence is given by u0 9 and un 2 3 un1 for n 1 b. Set the mode of the calculator to Seq instead of Func and enter the function as shown on the next page in Figure 1.2-4a. Figure 1.2-4b displays the table of values of the function, and Figure 1.2-4c displays the graph of the function. Section 1.2 Mathematical Patterns 17 10 0 0 Figure 1.2-4c Figure 1.2-4a Figure 1.2-4b c. As shown in Figures 1.2-4b and 1.2-4c, the height on the fourth bounce is approximately 1.7778 feet. 10 ■ Applications using Sequences Example 5 Salary Raise Sequence If the starting salary for a job is $20,000 and a raise of $2000 is earned at the end of each year of work, what will the salary be at the end of the sixth year? Find a recursive function to represent this problem and use a table and a graph to find the solution. Solution is 20,000. The amount of money earned at the end of The initial term, u1 the first year, u0,, will be 2000 more than un1 20,000 and un u0 The recursive function u0. 2000 for n 1 will generate the sequence that represents the salaries for each year. As shown in Figures 1.2-5a and 1.2-5 |
b, the salary at the end of the sixth year will be $32,000. 50,000 Figure 1.2-5a 0 0 Figure 1.2-5b 10 ■ In the previous examples, the recursive formulas were obtained by either adding a constant value to the previous term or by multiplying the previous term by a constant value. Recursive functions can also be obtained by adding different values that form a pattern. 18 Chapter 1 Number Patterns Example 6 Sequence Formed by Adding a Pattern of Values A chord is a line segment joining two points of a circle. The following diagram illustrates the maximum number of regions that can be formed by 1, 2, 3, and 4 chords, where the regions are not required to have equal areas. 1 Chord 2 Chords 3 Chords 4 Chords 2 Regions 4 Regions 7 Regions 11 Regions a. Find a recursive function to represent the maximum number of regions formed with n chords. b. Use a table to find the maximum number of regions formed with 20 chords. Solution 300 0 0 Figure 1.2-6a Let the initial number of regions occur with 1 chord, so. The maximum number of regions formed for each number or chords is shown in the following table. u1 2 Number of chords 1 2 3 4 p n Maximum number of regions u1 u2 u3 u4 un 2 4 u1 7 u2 11 u3 p un1 n 2 3 4 21 Figure 1.2-6b The recursive function is shown as the last entry in the listing above, and the table and graph, as shown in Figures 1.2-6a and 1.2-6b, identify the 20th term of the sequence as 211. Therefore, the maximum number of regions that can be formed with 20 chords is 211. ■ Example 7 Adding Chlorine to a Pool Dr. Miller starts with 3.4 gallons of chlorine in his pool. Each day he adds 0.25 gallons of chlorine and 15% evaporates. How much chlorine will be in his pool at the end of the sixth day? Solution The initial amount of chlorine is 3.4 gallons, so and each day 0.25 gallons of chlorine are added. Because 15% evaporates, 85% of the mixture remains. u0 3.4 Section 1.2 Mathematical Patterns 19 The amount of chlorine in the pool at the end of the first day is obtained by adding 0.25 to 3.4 and then multiplying the result by 0.85. 0.85 1 3 |
.4 0.25 2 3.1025 The procedure is repeated to yield the amount of chlorine in the pool at the end of the second day. Continuing with the same pattern, the recursive form for the sequence is 0.85 1 3.1025 0.25 2.85 2 u0 3.4 and un 0.85 un1 0.25 for n 1. 1 As shown in Figures 1.2-7a and 1.2-7b, approximately 2.165 gallons of chlorine will be in the pool at the end of the sixth day. 2 ■ Figure 1.2-7a 10 Figure 1.2-7b 4 0 0 Exercises 1.2 In Exercises 1 – 4, graph the first four terms of the sequence. 11. u1 un 1, u2 un1 2, u3 un2 3, un3 and for n 4 1. 2. 3. 4. 2, 5, 8, 11, p. 6 3, 6, 12, 24, p. 6 4, 5, 8, 13, p. 6 4, 12, 36, 108, p 5 5 5 5 6 In Exercises 5–8, define the sequence recursively and graph the sequence. 5 5 5 5. 6. 7. 8. 6, 4, 2, 0, 2, p 6 4, 8, 16, 32, 64, p 6 6, 11, 16, 21, 26, p 8, 4, 2, 1 12. u0 1, u1 1 and un nun1 for n 2 13. A really big rubber ball will rebound 80% of its height from which it is dropped. If the ball is dropped from 400 centimeters, how high will it bounce after the sixth bounce? 14. A tree in the Amazon rain forest grows an average of 2.3 cm per week. Write a sequence that represents the weekly height of the tree over the course of 1 year if it is 7 meters tall today. Write a recursive formula for the sequence and graph the sequence. 15. If two rays have a common endpoint, one angle is formed. If a third ray is added, three angles are formed. See the figure below. In Exercises 9–12, find the first five terms of the given sequence. 2 3 1 9. u1 4 and un 2un1 3 for n 2 10. u1 5 and un 1 3 un1 4 for n 2 Write a recursive formula |
for the number of angles formed with n rays if the same pattern continues. Graph the sequence. Use the formula to find the number of angles formed by 25 rays. 20 Chapter 1 Number Patterns 16. Swimming pool manufacturers recommend that the concentration of chlorine be kept between 1 and 2 parts per million (ppm). They also warn that if the concentration exceeds 3 ppm, swimmers experience burning eyes. If the concentration drops below 1 ppm, the water will become cloudy. If it drops below 0.5 ppm, algae will begin to grow. During a period of one day 15% of the chlorine present in the pool dissipates, mainly due to evaporation. a. If the chlorine content is currently 2.5 ppm and no additional chlorine is added, how long will it be before the water becomes cloudy? b. If the chlorine content is currently 2.5 ppm and 0.5 ppm of chlorine is added daily, what will the concentration eventually become? c. If the chlorine content is currently 2.5 ppm and 0.1 ppm of chlorine is added daily, what will the concentration eventually become? d. How much chlorine must be added daily for the chlorine level to stabilize at 1.8 ppm? 17. An auditorium has 12 seats in the front row. Each successive row, moving towards the back of the auditorium, has 2 additional seats. The last row has 80 seats. Write a recursive formula for the number of seats in the nth row and use the formula to find the number of seats in the 30th row. 18. In 1991, the annual dividends per share of a stock were approximately $17.50. The dividends were increasing by $5.50 each year. What were the approximate dividends per share in 1993, 1995, and 1998? Write a recursive formula to represent this sequence. 19. A computer company offers you a job with a starting salary of $30,000 and promises a 6% raise each year. Find a recursive formula to represent the sequence, and find your salary ten years from now. Graph the sequence. new students. What will be the enrollment 8 years from now? 22. Suppose you want to buy a new car and finance it by borrowing $7,000. The 12-month loan has an annual interest rate of 13.25%. a. Write a recursive formula that provides the declining balances of the loan for a monthly payment of $200. b. Write out the first five terms of this sequence. c. What is the unpaid balance after 12 months? |
d. Make the necessary adjustments to the monthly payment so that the loan can be paid off in 12 equal payments. What monthly payment is needed? 23. Suppose a flower nursery manages 50,000 flowers and each year sells 10% of the flowers and plants 4,000 new ones. Determine the number of flowers after 20 years and 35 years. 24. Find the first ten terms of a sequence whose first u1 two terms are for n 3 term 1 terms. 2 and 1 and whose nth is the sum of the two preceding 1 u2 Exercises 25–29 deal with prime numbers. A positive integer greater than 1 is prime if its only positive integer factors are itself and 1. For example, 7 is prime because its only factors are 7 and 1, but 15 is not prime because it has factors other than 15 and 1, namely, 3 and 5. 25. Critical Thinking a. Let un6 be the sequence of 5 prime integers in their usual ordering. Verify that the first ten terms are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. b. Find u17, u18, u19, u20. In Exercises 26 – 29, find the first five terms of the sequence. 26. Critical Thinking un is the nth prime integer larger 20. Book sales in the United States (in billions of than 10. dollars) were approximated at 15.2 in the year 1990. The book sales increased by 0.6 billion each year. Find a sequence to represent the book sales for the next four years, and write a recursive formula to represent the sequence. Graph the sequence and predict the number of book sales in 2003. 21. The enrollment at Tennessee State University is currently 35,000. Each year, the school will graduate 25% of its students and will enroll 6,500 27. Critical Thinking un is the square of the nth prime integer. 28. Critical Thinking un integers less than n. is the number of prime 29. Critical Thinking un is the largest prime integer less than 5n. Exercises 30–34 deal with the Fibonacci sequence { which is defined as follows: un } 1, u2 u1 preceding terms, and for un 1, n 3, un is the sum of the two That is, un1 un2 u1 u2 uz u3 u4 and so on. u3 u4 u5 u1 u2 u3 30. Critical Thinking Leonardo Fibonacci |
discovered the sequence in the thirteenth century in connection with the following problem: A rabbit colony begins with one pair of adult rabbits, one male and one female. Each adult pair produces one pair of babies, one male and one female, every month. Each pair of baby rabbits becomes adult and produces its first offspring at age two months. Assuming that no rabbits die, how many Section 1.3 Arithmetic Sequences 21 n 1, 2, 3, p? adult pairs of rabbits are in the colony at the end of n months, helpful to make up a chart listing for each month the number of adult pairs, the number of onemonth-old pairs, and the number of baby pairs. Hint: It may be 31. Critical Thinking List the first ten terms of the Fibonacci sequence. 32. Critical Thinking Verify that every positive integer less than or equal to 15 can be written as a Fibonacci number or as a sum of Fibonacci numbers, with none used more than once. 33. Critical Thinking Verify that perfect square for un2 n 1, 2, p, 10. 5 1 2 4 1 n 2 1 is a 34. Critical Thinking Verify that for n 2, 3, p, 10. 1 n1 1 2 2 un2 1 un1 un1 1.3 Arithmetic Sequences Objectives • Identify and graph an arithmetic sequence • Find a common difference • Write an arithmetic sequence recursively and explicitly • Use summation notation • Find the nth term and the nth partial sum of an arithmetic sequence An arithmetic sequence, which is sometimes called an arithmetic progression, is a sequence in which the difference between each term and the preceding term is always constant. Example 1 Arithmetic Sequence Are the following sequences arithmetic? If so, what is the difference between each term and the term preceding it? a. b. 5 5 14, 10, 6, 2,2, 6, 10, p 3, 5, 8, 12, 17, p 6 6 Solution a. The difference between each term and the preceding term is 4. So this is an arithmetic sequence with a difference of 4. b. The difference between the 1st and 2nd terms is 2 and the difference between the 2nd and 3rd terms is 3. The differences are not constant, therefore this is not an arithmetic sequence. ■ is an arithmetic sequence, then for each un un1 n 2, the term preceding is some constant—usually called un6 5 is If un un1 d. |
Therefore, un and the difference d. un1 22 Chapter 1 Number Patterns Recursive Form of an Arithmetic Sequence In an arithmetic sequence { un } un1 for some constant d and all n 2. un d The number d is called the common difference of the arithmetic sequence. Example 2 Graph of an Arithmetic Sequence is an arithmetic sequence with un6 If two terms, 5 u1 3 and u2 4.5 as its first 15 0 0 Figure 1.3-1a a. find the common difference. b. write the sequence as a recursive function. c. give the first seven terms of the sequence. d. graph the sequence. Solution a. The sequence is arithmetic and has a common difference of u1 4.5 3 1.5 u2 10 Figure 1.3-1b b. The recursive function that describes the sequence is n 2 1.5 3 and un1 for un u1 c. The first seven terms are 3, 4.5, 6, 7.5, 9, 10.5, and 12, as shown in Figure 1.3-1a. d. The graph of the sequence is shown in Figure 1.3-1b. ■ Explicit Form of an Arithmetic Sequence Example 2 illustrated an arithmetic sequence expressed in recursive form in which a term is found by using preceding terms. Arithmetic sequences can also be expressed in a form in which a term of the sequence can be found based on its position in the sequence. Example 3 Explicit Form of an Arithmetic Sequence Confirm that the sequence un expressed as 7 un n 1 1 2 un1 4. 4 with u1 7 can also be Solution Use the recursive function to find the first few terms of the sequence. u1 u2 7 7 4 3 Section 1.3 Arithmetic Sequences 23 2 u3 u4 u5 1 1 1 is 12 5 4 7 4 4 7 16, which is the first term of the sequence with Notice that the common difference of 4 added twice. Also, which is is the first term of the sequence with the common difference of 4 added n 1 The three times. Because this pattern continues, 2 table in Figure 1.3-2b confirms the equality of the two functions. n 1 7 and un 4 with u1 7 3 4, 7 7 un1 4. 4 un un u3 u4 1 1 2 ■ Figure 1.3-2a As shown in Example 3, if un1 difference d, then for each tion in terms of n |
, the position of the term. un6 5 n 2, un is an arithmetic sequence with common can be written as a func- d Figure 1.3-2b Applying the procedure shown in Example 3 to the general case shows that u2 u3 u4 u5 u1 u2 u3 u4 d d d d u1 u1 u1 1 1 1 d 2 2d 3d d u1 d u1 d u1 2d 3d 4d 2 2 u5 Notice that 4d is added to u1 So ence, d, added un n 1 yields un. times. 1 2 u1 to obtain is the sum of n numbers:. In general, adding to and the common differ- d 1 2 u1 n 1 Explicit Form of an Arithmetic Sequence In an arithmetic sequence { un n 1 u1 un } with common difference d, d for every n 1. 1 2 u0, If the initial term of a sequence is denoted as arithmetic sequence with common difference d is the explicit form of an un u0 nd for every n 0. Example 4 Explicit Form of an Arithmetic Sequence Find the nth term of an arithmetic sequence with first term mon difference of 3. Sketch a graph of the sequence. 5 and com- Solution Because u1 10 5 and u1 d 3, n 1 1 2 un the formula in the box states that n 1 3 3n 8 d 5 1 2 Figure 1.3-3 The graph of the sequence is shown in Figure 1.3-3. ■ 30 0 10 24 Chapter 1 Number Patterns Example 5 Finding a Term of an Arithmetic Sequence What is the 45th term of the arithmetic sequence whose first three terms are 5, 9, and 13? Solution The first three terms show that d, is 4. Apply the formula with and that the common difference, u45 u1 1 45 1 2 44 4 2 21 1 181 ■ 5 u1 n 45. d 5 Example 6 Finding Explicit and Recursive Formulas un6 If is an arithmetic sequence with 5 sive formula, and an explicit formula for u6 57 un. and u10 93, find u1, a recur- Solution The sequence can be written as p, 57, —, —, —, 93, p u8 u9 u10 u7 { { { { { u6 The common difference, d, can be found by the difference between 93 and 57 divided by the number of times d must be added to |
57 to produce 93 (i.e., the number of terms from 6 to 10). d 93 57 10 6 36 4 9 d 9 Note that is the difference of the output values (terms of the sequence) divided by the difference of the input values (position of the terms of the sequence), which represents the change in output per unit change in input. The value of equation u1 can be found by using n 6, u6 57 and d 9 in the n 1 6 1 d 2 9 2 57 5 9 57 45 12 1 1 u1 u6 57 u1 u1 d 9, and the recursive form of the arithmetic sequence u1 Because is given by 12 u1 12 and un un1 9, for n 2 The explicit form of the arithmetic sequence is given by un n 1 12 1 9n 3, for n 1. 9 2 ■ NOTE If u1, u2, u3, p is an arithmetic sequence, then an expression of u3 u1 the form (sometimes written as q u2 p Section 1.3 Arithmetic Sequences 25 Summation Notation It is sometimes necessary to find the sum of various terms in a sequence. For instance, we might want to find the sum of the first nine terms of the sequence Mathematicians often use the capital Greek letter sigma to abbreviate such a sum as follows. un6 5. 1 2 un ) is called an a n1 arithmetic series. 9 a i1 ui u1 u2 u3 u4 u5 u6 u7 u8 u9 Similarly, for any positive integer m and numbers c1, c2, p, cm, Summation Notation m a k1 means c1 ck c2 c3 p cm Example 7 Sum of a Sequence Compute each sum. a. 5 a n1 1 7 3n 2 b. 4 a n1 3 3 n 1 1 4 2 4 Solution a. Substitute 1, 2, 3, 4, and 5 for n in the expression 7 3n and add the terms. 5 a n1 1 7 3n 10 12 1 2 7 15 2 b. Substitute 1, 2, 3, and 4 for n in the expression add the terms. 3 n 1 2 1 4, and 4 a n1 12 12 2 3 4 3 3 4 4 ■ 26 Chapter 1 Number Patterns Technology Tip SEQ is in the OPS submenu of the TI LIST menu and in the LIST submenu of the Casio OPTN |
menu. SUM is in the MATH submenu of the TI LIST menu. SUM is in the LIST submenu of the Casio OPTN menu. Using Calculators to Compute Sequences and Sums Calculators can aid in computing sequences and sums of sequences. The SEQ (or MAKELIST) feature on most calculators has the following syntax. SEQ(expression, variable, begin, end, increment) The last parameter, increment, is usually optional. When omitted, increment defaults to 1. Refer to the Technology Tip about which menus contain SEQ and SUM for different calculators. The syntax for the SUM (or LIST ) feature is SUM(list[, start, end]), where start and end are optional. When start and end are omitted, the sum of the entire list is given. Combining the two features of SUM and SEQ can produce sums of sequences. SUM(SEQ(expression, variable, begin, end)) Example 8 Calculator Computation of a Sum Use a calculator to display the first 8 terms of the sequence un 7 3n and to compute the sum 50 a n1 7 3n. Solution Using the Technology Tip, enter which produces Fig1 ure 1.3-4. Additional terms can be viewed by using the right arrow key to scroll the display, as shown at right below., 2 SEQ 7 3n, n, 1, 8 Figure 1.3-4 Figure 1.3-5 fore, 1 22 7 3n 3475. 1 50 a n1 17. The first 8 terms of the sequence are To compute the sum of the first 50 terms of the sequence, enter SUM Figure 1.3-5 shows the resulting display. There- 4, 1, 2, 5, 8, 11, 14, 7 3n, n, 1, 50 SEQ and. Partial Sums Suppose k terms of the sequence is called the kth partial sum of the sequence. is a sequence and k is a positive integer. The sum of the first un6 5 ■ Section 1.3 Arithmetic Sequences 27 Calculator Exploration Write the sum of the first 100 counting numbers. Then find a pattern to help find the sum by developing a formula using the terms in the sequence. Partial Sums of an Arithmetic Sequence un If { } is an arithmetic sequence with common difference d, then for each positive integer k, the kth partial sum can be found by using either of the following formulas. 1. k a n1 k 2. |
a n1 un k 2 (u1 un ku1 uk) k(k 1) 2 d Sk represent the kth partial sum Proof Let terms of the arithmetic sequence in two ways. In the first representation of Write the u1 Sk, u2 p uk. repeatedly add d to the first term. u3 p uk2 u2 d 2d u1 u1 u1 u1 Sk uk1 uk p 3 4 In the second representation of uk uk uk1 uk Sk 3 3, repeatedly subtract d from the kth term. 2 1 4 4 Sk u1 k 1 d uk2 p u3 d 2d uk u2 p u1 uk k 1 d 4 Sk 3 3 are added, the multiples of d add to zero 1 2 4 4 If the two representations of and the following representation of u1 uk d d u1 uk u1 uk Sk Sk 4 4 3 3 2Sk 2d 2d is obtained. p p 4 4 u1 3 uk terms ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ 2Sk Sk u1 1 k 2 1 u1 1 uk2 uk2 uk2 u1 p u1 1 uk2 k u1 1 uk2 Divide by 2. The second formula is obtained by letting equation. uk u1 k 1 2 1 d in the last k 2 3 u1 u1 k 1 1 k 2 3 d 2 4 2u1 k 1 ■ d 2 4 1 Sk k 2 1 ku1 u1 uk2 k 1 k 1 2 2 d 28 Chapter 1 Number Patterns Example 9 Partial Sum of a Sequence Find the 12th partial sum of the arithmetic sequence below. 8, 3, 2, 7, p Solution First note that d, the common difference, is 5 and u1 8. u12 u1 8 11 12 1 5 2 d 47 1 1 2 Using formula 1 from the box on page 27 yields the 12th partial sum. 12 a n1 un 12 2 1 8 47 234 2 ■ Example 10 Partial Sum of a Sequence Find the sum of all multiples of 3 from 3 to 333. Solution Note that the desired sum is the partial sum of the |
arithmetic sequence 3, 6, 9, 12, p. The sequence can be written in the form 3 1, 3 2, 3 3, 3 4, p, 333 3 111 is the 111th term. The 111th partial sum of the where sequence can be found by using formula 1 from the box on page 27 with k 111, 333. 3, and u111 u1 111 a n1 un 111 2 1 3 333 111 2 1 2 336 2 18,648 ■ Example 11 Application of Partial Sums If the starting salary for a job is $20,000 and you get a $2000 raise at the beginning of each subsequent year, how much will you earn during the first ten years? Solution The yearly salary rates form an arithmetic sequence. 20,000 22,000 24,000 26,000 p u1 The tenth-year salary is found using 10 1 u1 2 d 1 20,000 9 1 2000 u10 $38,000 2 20,000 and d 2000. Section 1.3 Arithmetic Sequences 29 The ten-year total earnings are the tenth partial sum of the sequence. 10 a n1 un 10 2 ˛ 1 u1 u102 20,000 38,000 10 2 ˛ 1 5˛1 58,000 2 $290,000 2 ■ Exercises 1.3 In Exercises 1–6, the first term, and the common difference, d, of an arithmetic sequence are given. Find the fifth term, the explicit form for the nth term, and sketch the graph of each sequence. u1, 1. u1 5, d 2 2. u1 4, d 5 3. u1 4, d 1 4 4. u1 6, d 2 3 5. u1 10, d 1 2 6. u1 p, d 1 5 In Exercises 7– 12, find the sum. 7. 9. 5 a i1 3i 16 a n1 1 2n 3 2 11. 36 a n151 2n 8 2 8. 4 a i1 1 2i 10. 12. 75 a n1 1 31 a n1 1 3n 1 2 300 n 1 1 2 ˛2 2 In Exercises 13–18, find the kth partial sum of the arithmetic sequence { } with common difference d. un 13. k 6, u1 2, d 5 14. k 8, u1 2 3, d 4 3 15. k 7, u |
1 3 4, d 1 2 16. k 9, u1 6, u9 24 17. k 6, u1 4, u6 14 18. k 10, u1 0, u10 30 In Exercises 19–24, show that the sequence is arithmetic and find its common difference. 19. 5 3 2n 6 21. 23. 24. 5 3n 2 e f c 2n 6 2b 3nc 5 5 20. 4 n e 3 f 22. p n 2 e f c constant 2 1 6 1 b, c constants 2 In Exercises 25–30, use the given information about the arithmetic sequence with common difference d to find un. and a formula for u5 25. u4 12, d 2 26. u7 8, d 3 27. u2 4, u6 32 28. u7 6, u12 4 29. u5 0, u9 6 30. u5 3, u9 18 In Exercises 31–34, find the sum. 31. 33. 20 a n1 1 3n 4 2 40 a n1 n 3 6 32. 34. 25 a n1 a n 4 5 b 30 a n1 4 6n 3 35. Find the sum of all the even integers from 2 to 100. 36. Find the sum of all the integer multiples of 7 from 7 to 700. 37. Find the sum of the first 200 positive integers. 38. Find the sum of the positive integers from 101 to 200 (inclusive). Hint: Recall the sum from 1 to 100. Use it and Exercise 37. 30 Chapter 1 Number Patterns 39. A business makes a $10,000 profit during its first year. If the yearly profit increases by $7500 in each subsequent year, what will the profit be in the tenth year? What will be the total profit for the first ten years? 40. If a man’s starting annual salary is $15,000 and he receives a $1000 increase to his annual salary every six months, what will he earn during the last six months of the sixth year? How much will he earn during the first six years? 41. A lecture hall has 6 seats in the first row, 8 in the 42. A monument is constructed by laying a row of 60 bricks at ground level. A second row, with two fewer bricks, is centered on that; a third row, with two fewer bricks, is centered on the second; and so on. The |
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