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ombus. Properties of a Square 1. A square has all the properties of a rectangle. 2. A square has all the properties of a rhombus. Methods of Proving That a Quadrilateral Is a Square We prove that a quadrilateral is a square by showing that it has the special properties of a square. 14365C10.pgs 7/10/07 8:50 AM Page 400... |
2. Raphael said that a square could be defined as a quadrilateral that is both equiangular and equilateral. Do you agree with Raphael? Justify your answer. Developing Skills In 3–6, the diagonals of square ABCD intersect at M. AC is the perpendicular bisector of 3. Prove that 4. If AC 3x 2 and BD 7x 10, find AC, BD, A... |
Prove that the diagonals of a square divide the square into four congruent isosceles right triangles. 15. Two line segments, AEC and BED, are congruent. Each is the perpendicular bisector of the other. Prove that ABCD is a square. 16. Prove that if the midpoints of the sides of a square are joined in order, another sq... |
oid in which the nonparallel sides are congruent. T S Q R TS QR If and QT > RS, then QRST is an isosceles trapezoid. The angles whose vertices are the endpoints of a base are called base angles. Here, Q and R are one pair of base angles because Q and R are endpoints of base. Also, T and S are a second pair of base angl... |
lower base angles and S and T as the upper base angles. The proof of this theorem for S and T is left to the student. (See exercise 15.) Theorem 10.20b If the base angles of a trapezoid are congruent, then the trapezoid is isosceles. Given Trapezoid QRST with QR ST and Q R T S Prove QT > RS Strategy Draw SP TQ. Prove ... |
a and 10.21b can also be written as a biconditional. Theorem 10.21 A trapezoid is isosceles if and only if the diagonals are congruent. Recall that the median of a triangle is a line segment from a vertex to the midpoint of the opposite sides. A triangle has three medians. A trapezoid has only one median, and it joins ... |
) y D(d, e) C(c, e) M N A(0, 0) B(b, 0) x Strategy Use a coordinate proof. Let the coordinates of A, B, C, D, M, and N be those used in the proof of Theorem 10.22. The length of a horizontal line segment is the absolute value of the difference of the x-coordinates of the endpoints. The proof of this theorem is left to ... |
and AD BC are parallel. The quadrilat- trapezoid is isosceles. d. The congruent legs of the trapezoid are AB CD AB and. CD x 3x 8 2x 8 x 4 AB x 4 BC 2x 1 CD 3x 8 2(4) 1 8 1 7 3(4) 8 12 8 4 DA x 1 4 1 5 14541C10.pgs 1/25/08 3:47 PM Page 408 408 Quadrilaterals Exercises Writing About Mathematics 1. Can a trapezoid have ... |
and R, are con- gruent. Use this fact to prove that the upper base angles of QRST, S and T, are congruent. 16. Prove Theorem 10.20b, “If the base angles of a trapezoid are congruent, then the trapezoid is isosceles.” 17. a. Prove Theorem 10.21b, “If the diagonals of a trapezoid are congruent, then the trapezoid is iso... |
length of the altitude, a line segment perpendicular to the base. Area of ABCD (AB)(BC) bh The formula for the area of every other polygon can be derived from this formula. In order to derive the formulas for the areas of other polygons from the formula for the area of a rectangle, we will use the following postulate.... |
Through C, draw a line parallel to g AB, and through B, draw a line parallel g AC to. Let the point of intersection of these lines be D. b. Prove that ABC DBC. c. Let E be a point on g AB such that g CE'AB of Example 1 to prove that the area of ABC bh. 1 2. Let AB b and CE h. Use the results 14365C10.pgs 7/10/07 8:50 ... |
. What is the area of ABCD? e. Let E and F be the coordinates of the fixed points under the reflection in the y-axis. Prove that AEAF is a square. f. What is the area of AEAF? 12. KM is a diagonal of parallelogram KLMN. The area of KLM is 94.5 square inches. a. What is the area of parallelogram KLMN? b. If MN 21.0 inch... |
d. 3. Find the area of ABCD in terms of d. 4. Let AB s. Express the area of ABCD in terms of s. 5. Write an equation that expresses the relationship between d and s. 6. Solve the equation that you wrote in step 5 for d in terms of s. 7. Use the result of step 6 to express the length of the hypotenuse of an isosceles r... |
is a parallelogram. 10.5 10.6 10.7 10.8 All angles of a rectangle are right angles. 10.9 The diagonals of a rectangle are congruent. 10.10 If a quadrilateral is equiangular, then it is a rectangle. 10.11 If the diagonals of a parallelogram are congruent, the parallelogram is a rectangle. 10.12 All sides of a rhombus a... |
• Upper base angles • Median of a trapezoid 10-8 Area of a polygon REVIEW EXERCISES 1. Is it possible to draw a parallelogram that has only one right angle? Explain why or why not. 2. The measure of two consecutive angles of a parallelogram are represented by 3x and 5x 12. Find the measure of each angle of the paralle... |
on graph paper and draw the quadrilateral. b. What kind of quadrilateral is ABCD? Justify your answer. c. Find the area of quadrilateral ABCD. 13. The vertices of quadrilateral DEFG are D(1, 1), E(4, 1), F(1, 3), and G(2, 1). a. Is the quadrilateral a parallelogram? Justify your answer. b. Is the quadrilateral a rhomb... |
the perpendicular bisectors of two adjacent sides of each of the following quadrilaterals. b. Construct the third and fourth perpendicular bisectors of the sides of each of the quadrilaterals. For which of the above quadrilaterals is the intersection of the first two perpendicular bisectors the same point as the inter... |
two sides of a triangle are 8 and 12. The measure of the third side cannot be (1) 16 (2) 12 (3) 8 (4) 4 7. The line segment is the median and the altitude of ABC. Which of BD the following statements must be false? (1) (2) BDA is a right triangle. (3) mA 90 (4) B is equidistant from A and C. 8. What is the equation of... |
that AO > CD. ABCD is a trapezoid with OD of a circle are congruent.) Part IV Answer all questions in this part. Each correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numeri... |
But the world around us is three-dimensional. The geometry of three dimensions is called solid geometry. We begin this study with some postulates that we can accept as true based on our observations. We know that two points determine a line. How many points are needed to determine a plane? A table or chair that has fo... |
line m are on the plane, then all of the points of m are on the plane. Therefore, there is exactly one plane that contains the given intersecting lines. The definition of parallel lines gives us another set of points that must lie in a plane. DEFINITION Parallel lines in space are lines in the same plane that have no ... |
. g AD g BC. Prove that A, B, C, and D must lie in a plane and form a parallelo- BED AEC C, and D must lie in a plane and form a square. and each segment is the perpendicular bisector of the other. Prove that A, B, 4. 5. In 6–9, use the diagram at the right. 6. Name two pairs of intersecting lines. 7. Name two pairs of... |
If two planes intersect, then they intersect in exactly one line. The Angle Formed by Two Intersecting Planes Fold a piece of paper. The part of the paper on one side of the crease represents a half-plane and the crease represents the edge of the half-plane. The folded paper forms a dihedral angle. 14365C11.pgs 7/31/0... |
the following theorem. Theorem 11.3 If a line not in a plane intersects the plane, then it intersects in exactly one point. 14365C11.pgs 7/12/07 1:05 PM Page 425 Perpendicular Lines and Planes 425 Given Line l is not in plane p and l intersects p. Prove Line l intersects p in exactly one point. Proof Use an indirect p... |
426 The Geometry of Three Dimensions Theorem 11.4 If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by these lines. Given A plane p determined by g AP and g BP, two lines that intersect at P. Line l such that g g l'AP l'B... |
, are each perpendicular to, and in plane q, construct g AD g AB g AC and g'AB g AC. Therefore,. Since p and q from a g'AD. Two lines in plane g AD. Similarly, g AB and g AD, are each perpendicular to. Therefore,'p g AC two lines in plane q, g AC'q. 14365C11.pgs 7/12/07 1:05 PM Page 428 428 The Geometry of Three Dimens... |
AD and plane q, given plane, there is only one line perpendicular to a given line at a given point. Our assumption is false, and there is only one line perpendicular to a given plane at a given point.. But in a Perpendicular Lines and Planes 429 q p B A C D As we noted above, in space, there are infinitely many lines ... |
perpendicular to a given line. Therefore, g AD g AC that is, C is on. Since section of planes p and q, and g AD are the same line, g AD g AC is the inter- is in plane p. Theorem 11.9 If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane. Given Plane p with on p... |
l. A p Proof Use an indirect proof. g AB Let and g AC be two lines through A in g l'AC g l'AB p. Assume that Therefore, l ⊥ p because if a line is perpendicular to each of two lines at their point of intersection, then the line is perpendicular to the plane determined by and. these lines. But it is given that l is not... |
. 14. Prove step 5 of Theorem 11.4. 15. Prove that if a line segment is perpendicular to a plane at the midpoint of the line segment, then every point in the plane is equidistant from the endpoints of the line segment. Given: ⊥ plane p at M, the midpoint of AB point in plane p. AB, and R is any p Prove: AR BR R R p M M... |
EXAMPLE 1 Plane p intersects plane q in g CD. Prove that if g AB g AB and and intersect, then planes q and r are plane r in g CD not parallel. Proof Let E be the point at which g AB and g intersect. Then E is a point on q CD and E is a point on r. Therefore, planes q and r intersect in at least one point and are not p... |
g LA. If one of two parallel lines is perpendicular to a third line, then the other is perpendicu⊥ lar to the third line, that is, since g'AB, then g NB g AB g LA. (3) Draw g BD g'AB in p. Because p and q form a right dihedral angle, NBD is a right angle, and so g NB g'BD. (4) Therefore, g NB is perpendicular to two l... |
determined by A, B, and C intersecting q at g AC ⊥ plane p. Therefore, are parallel, g AB g BE g BE Then, in plane r, g AB g'BE.. Since p and q. It is given that g'AC g AB. is per- C p E q A B r 14365C11.pgs 7/12/07 1:05 PM Page 437 (2) Let D be a point in p. Let s be the plane determined by A, B, and D. Since p and q... |
perpendicular to plane p and line l is not perpendicular to plane q. Is p q? Solution Assume that p q. If two planes are parallel, then a line perpendicular to one is perpendicular to the other. Therefore, since l is perpendicular to plane p, l must be perpendicular to plane q. This contradicts the given statement tha... |
q p through point D on AB 11. Plane p is perpendicular to g PQ Prove that AQ > BQ. at Q and two points in p, A and B, are equidistant from P. 14365C11.pgs 7/12/07 1:06 PM Page 440 440 The Geometry of Three Dimensions 12. Noah is building a tool shed. He has a rectangular floor in place and wants to be sure that the po... |
of these parallelograms, the lateral edges, are also congruent and parallel. Therefore, we can make the following statement: The lateral edges of a prism are congruent and parallel. DEFINITION A right prism is a prism in which the lateral sides are all perpendicular to the bases. All of the lateral sides of a right pr... |
Dimensions DEFINITION A parallelepiped is a prism that has parallelograms as bases. Examples of parallelepipeds Rectangular Solids DEFINITION A rectangular parallelepiped is a parallelepiped that has rectangular bases and lateral edges perpendicular to the bases. A rectangular parallelepiped is usually called a rectan... |
inches and the height of the prism is 5 inches. a. How many lateral sides does this prism have and what is their shape? b. What is the lateral area of the prism? Solution a. Because this is a prism with a triangular base, the prism has three lateral sides. Because it is a right prism, the lateral sides are rectangles.... |
angles. a. Is an altitude of the solid congruent to an altitude of one of the rectangular faces? Explain your answer. b. Is an altitude of the solid congruent to an altitude of one of the faces that are parallel- ograms? Explain your answer. Developing Skills In 3–6, find the surface area of each of the rectangular sol... |
through B perpendicular to the plane of ABC intersects the plane of DEF at E. a. Prove that the lateral faces of the prism are rectangles. b. When are the lateral faces of the prism congruent polygons? Justify your answer. 14. A right prism has bases that are squares. The area of one base is 81 square feet. The latera... |
is a unit of volume called a cubic centimeter. If the bases of a rectangular solid measure 8 centimeters by 5 centimeters, we know that the area of a base is 8 5 or 40 square centimeters and that 40 cubes each with a volume of 1 cubic centimeter can fill one base. If the height of the solid is 3 centimeters, we know t... |
when describing a prism. For example, each of the congruent polygons in parallel planes is a base of the prism. The distance between the parallel planes is the height of the prism. In order to find the area of a base that is a triangle or a parallelogram, we use the length of a base and the height of the triangle or p... |
.6 feet. 5. One base is a right triangle whose legs measure 5 inches and 7 inches. The height of the prism is 9 inches. 6. One base is a square whose sides measure 12 centimeters and the height of an altitude is 75 millimeters. 7. One base is parallelogram ABCD and the other is parallelogram ABCD, AB 47 cm, AB AAr'AB, ... |
whose altitude is perpendicular to the base at its center. The lateral edges of a regular polygon are congruent. Therefore, the lateral faces of a regular pyramid are isosceles triangles. The length of the altitude of a triangular lateral face of a regular pyramid, PB, is the slant height of the pyramid. slant height ... |
and the altitude is perpendicular to the base at its center. The center of a regular polygon is defined as the point that is equidistant to its vertices. In a regular polygon with three sides, an equilateral triangle, we proved that the perpendicular bisector of the sides of the triangle meet in a point that is equidi... |
. AB V Solution The slant height of the pyramid is the height of a lat- E eral face. Therefore: Area of nABV 5 1 5 1 5 15 2bh 2(2.5)(6) 2 cm2 F D 6 c m C A 2.5 cm B The lateral faces of the regular pyramid are congruent. Therefore, they have equal areas. There are six lateral faces. Lateral area of the pyramid 5 6 15 2... |
side of a triangle is 10.7 centimeters, the slant height is 9.27 centimeters, and the height of the prism is 8.74 centimeters. a. Find the area of the base of the tetrahedron. b. Find the lateral area of the tetrahedron. c. Find the total surface area of the tetrahedron. d. Find the volume of the tetrahedron. 12. When... |
points on the bases. The height of a cylinder is the length of an altitude. lateral surface altitude base The most common cylinder is one that has bases that are congruent circles. This cylinder is a circular cylinder. If the line segment joining the centers of the circular bases is perpendicular to the bases, the cyli... |
06 PM Page 455 Express this result as a rational approximation rounded to the nearest cubic centimeter. 504p 1,583.362697 1,584 cm3 Cylinders 455 Answers a. 754 cm2 b. 1,584 cm3 Exercises Writing About Mathematics 1. Amy said that if the radius of a circular cylinder were doubled and the height decreased by one-half, t... |
nearest tenth? 12. A truck that delivers gasoline has a circular cylindrical storage space. The diameter of the bases of the cylinder is 11 feet, and the length (the height of the cylinder) is 17 feet. How many whole gallons of gasoline does the truck hold? (Use 1 cubic foot 7.5 gallons.) 13. Karen makes pottery on a ... |
Q is the height of the cone, and PQ is the slant height of the cone. OQ Q O P p 14365C11.pgs 7/12/07 1:06 PM Page 457 Cones 457 We can make a model of a right circular cone. Draw a large circle on a piece of paper and draw two radii. Cut out the circle and remove the part of the circle between the two radii. Join the t... |
in.2 Answer 14365C11.pgs 7/12/07 1:06 PM Page 458 458 The Geometry of Three Dimensions b. Area of the base 5 p(10)2 5 100p cm2 Answer c. Total surface area 5 260p 1100p 5 360p in.2 Answer 26 in. 24 in. 10 in. EXAMPLE 2 A cone and a cylinder have equal volumes and equal heights. If the radius of the base of the cone is... |
volume of a cone is 56 cubic centimeters and the area of the base is 48 square centime- ters. What is the height of the cone to the nearest tenth? 9. The area of the base of a cone is equal to the area of the base of a cylinder, and their volumes are equal. If the height of the cylinder is 2 feet, what is the height o... |
a sphere is greater than the radius of the sphere, the plane will have no points in common with the sphere. If the distance of a plane from the center of a sphere is equal to the radius of the sphere, the plane will have one point in common with the sphere. If the distance of a plane from the center of a sphere is les... |
plane equidistant from a fixed point. We can write Theorems 11.14a and 11.14b as a single theorem. Theorem 11.14 The intersection of a plane and a sphere is a circle. In the proof of Theorem 11.14b, we drew right triangle OAC with OA the radius of the sphere and AC the radius of the circle at which the plane and the s... |
of four great circles. Let S be the surface area of a sphere of radius r. Then the surface area of the sphere is: S 4pr 2 The volume of a sphere is equal to four-thirds the product of p and the cube of the radius. Let V be the volume of a sphere of radius r. Then the volume of the sphere is: V 4 3pr 3 Find the surface... |
many whole cups of water will come closest to filling the vase? (1 cup 14.4 cubic inches) 12. The radius of a ball is 5.0 inches. The ball is made of a soft foam that weighs 1 ounce per 40 cubic inches. How much does the ball weigh to the nearest tenth? 14365C11.pgs 8/2/07 5:54 PM Page 464 464 The Geometry of Three Di... |
the line is perpendicular to the plane. • Parallel planes are planes that have no points in common. • A line is parallel to a plane if it has no points in common with the plane. • The distance between two planes is the length of the line segment perpen- dicular to both planes with an endpoint on each plane. • A polyhe... |
segment perpendicular to a plane at O and A be a point on a circle in the plane with center at O. A right circular cone is the solid figure that is the union of a circular base and the surface generated by line segment as P moves around the circle. AP • A sphere is the set of all points equidistant from a fixed point ... |
are parallel. 11.13 Parallel planes are everywhere equidistant. 14541C11.pgs 1/25/08 3:53 PM Page 467 Vocabulary 467 11.14 The intersection of a plane and a sphere is a circle. 11.14a A great circle is the largest circle that can be drawn on a sphere. 11.15 If two planes are equidistant from the center of a sphere and... |
Geometry of Three Dimensions 11-8 Right circular conical surface • Right circular cone • Vertex of a cone • Base of a cone • Altitude of a cone • Height of a cone • Slant height of a cone • Frustum of a cone 11-9 Sphere • Center of a sphere • Radius of a sphere • Great circle of a sphere • Symmetry plane REVIEW EXERCI... |
15. Two planes intersect a sphere at equal distances from the center of the sphere. Are the circles at which the planes intersect the sphere congruent? 16. Plane p intersects a sphere 2 centimeters from the center of the sphere and plane q contains the center of the sphere and intersects the sphere. Are the circles at... |
container? 14365C11.pgs 8/2/07 5:55 PM Page 470 470 The Geometry of Three Dimensions Exploration A regular polyhedron is a solid, all of whose faces are congruent regular polygons with the sides of the same number of polygons meeting at each vertex. There are five regular polyhedra: a tetrahedron, a cube, an octahedro... |
(4, 2) (2) (4, 2) (3) (4, 2) (4) (2, 1) 4. The lengths of the diagonals of a rhombus are 8 centimeters and 12 cen- timeters. The area of the rhombus is (1) 24 cm2 (2) 32 cm2 (3) 48 cm2 (4) 96 cm2 5. Two parallel lines are cut by a transversal. The measure of one interior angle is x 7 and the measure of another interio... |
credit. 11. A leg, AB, of isosceles ABC is congruent to a leg, DE, of isosceles DEF. The vertex angle, B, of isosceles ABC is congruent to the vertex angle, E, of isosceles DEF. Prove that ABC DEF. 12. In triangle ABC, altitude CD bisects C. Prove that the triangle is isosceles. Part III Answer all questions in this p... |
The Distance Formula Chapter Summary Vocabulary Review Exercises Cumulative Review 474 RATIO, PROPORTION, AND SIMILARITY The relationship that we know as the Pythagorean Theorem was known by philosophers and mathematicians before the time of Pythagoras (c. 582–507 B.C.). The Indian mathematician Baudha¯yana discovered... |
AB 20 millimeters and DE 10 millimelengths by ters. We can compare these means of a ratio, 20 10 or 20 : 10. Since a ratio, like a fraction, is a comparison of two numbers by division, a ratio can be simplified by dividing each term of the ratio by a common factor. Therefore, the ratio of AB to DE can be written as 10... |
the ratio 3 : 4, we may write 16 5 3 12 4. The equa- is called a proportion. The proportion can also be written as 16 5 3 12 tion 4 12 : 16 3 : 4. DEFINITION A proportion is an equation that states that two ratios are equal. b 5 c a d The proportion can be written also as a : b c : d. The four numbers a, b, c, and d a... |
2 3 14365C12.pgs 7/10/07 8:56 AM Page 478 478 Ratio, Proportion, and Similarity The four proportions at the bottom of page 477 demonstrate the following corollary: Corollary 12.1c If the products of two pairs of factors are equal, the factors of one pair can be the means and the factors of the other the extremes of a ... |
. 3. 6 : 15, 4 : 10 6. 10 : 15, 8 : 12 4. 8 : 7, 56 : 49 7. 9 : 3, 16 : 4 5. 49 : 7, 1 : 7 8. 3a : 5a, 12 : 20 (a 0) In 9–11, use each set of numbers to form two proportions. 9. 30, 6, 5, 1 10. 18, 12, 6, 4 11. 3, 10, 15, 2 12. Find the exact value of the geometric mean between 10 and 40. 13. Find the exact value of th... |
point of AC BC DE The line segment joining the midpoints of ABC forms a new triangle, • D is the midpoint of DEC. What are the ratios of the sides of these triangles? DC AC 5 1 2 EC BC 5 1. 2 1 and 2AB 1. Therefore, DC 2AC 1. Therefore, EC 2BC, it appears that DE • E is the midpoint of If we measure and and AC BC. AB A... |
A EDC and B DEC because AB DE that they are corresponding angles of parallel lines. We also know that C C. Therefore, for ABC and DEC, the corresponding angles are congruent and the ratios of the lengths of corresponding sides are equal. and C Again, in ABC, let D be the midpoint of and E be the midpoint of. AC DE Now... |
2. Statements BC 5 DE AB EF (AB)(EF) (BC)(DE) 3. (AB)(EF) (BC)(DE) (AB)(DE) (AB)(DE) 4. (AB)(EF DE) (DE)(BC AB) (AB)(DF) (DE)(AC) AC 5 DE AB DF 6. 5. A D B C E F Reasons 1. Given. 2. The product of the means equals the product of the extremes. 3. Addition postulate. 4. Distributive property. 5. Substitution postulate.... |
ABC and DEF proportionally? are line segments. If AB 10, AC 15, DE 8, and DF 12, Solution If AB 10 and AC 15, then: If DE 8 and DF 12, then: BC 5 15 2 10 5 5 AB : BC 5 10 : 5 5 2 : 1 EF 5 12 2 8 5 4 DE : EF 5 8 : 4 5 2 : 1 Since the ratios of AB : BC and DE : EF are equal, B and E divide DEF proportionally. ABC and 14... |
AM Page 485 In 12–15, the line segments PQ QR. ABC and PQR are divided proportionally by B and Q. AB BC and Proportions Involving Line Segments 485 12. Find PQ when AB 15, BC 25, and QR 35. 13. Find BC when AB 8, PQ 20, and PR 50. 14. Find AC when AB 12, QR 27, and BC PQ. 15. Find AB and BC when AC 21, PQ 14, and QR 3... |
pgs 7/10/07 8:56 AM Page 486 486 Ratio, Proportion, and Similarity 12-3 SIMILAR POLYGONS Two polygons that have the same shape but not the same size are called similar polygons. In the figure to the right, ABCDE PQRST. The symbol is read “is similar to.” These polygons have the same shape because their corresponding an... |
B B BC : BrCr 5 14 : 7 C C CA : CrAr 5 22 : 11 The ratio of similitude for the triangles is 2 : 1. Equivalence Relation of Similarity The relation “is similar to” is true for polygons when their corresponding angles are congruent and their corresponding sides are in proportion. Thus, for a given set of triangles, we c... |
of corresponding sides of two congruent polygons? 4. Are all congruent polygons similar? Explain your answer. 5. Are all similar polygons congruent? Explain your answer. 6. What must be the constant of proportionality of two similar polygons in order for the poly- gons to be congruent? 7. The sides of a triangle measu... |
, ABC. AB 5 3 DE STEP 2. Draw any line 1 STEP 3. Construct GDE A and HED B. Let F be the intersection, that is, DE 3AB. h h. DG EH a. Find the measures of AC 5 3 CB 5 3 EF DF b. Is 1 1 c. Is DEF ABC? d. Repeat this construction using a different ratio of similitude. Are the trian-, and and? Is DF AC EF BC of?,,. gles s... |
CA CrAr. C Prove ABC ABC Proof We will construct a third triangle DEC that is similar to both ABC and ABC. By the transitive property of similarity, we can conclude that ABC ABC Let AC AC. Choose point D on so that DC AC. Choose point BrCr ArCr E on gruent, so CDE A and C C. Therefore, ABC DEC by AA. If two polygons a... |
and ABC DFC C E G D A F B Proof We are given ADEC and AD DE EC. Then AC AD DE EC. By the substitution postulate, AC AD AD AD 3AD and DC DE EC AD AD 2AD. BFGC and BF FG GC. Then BC BF FG GC. We are also given By the substitution postulate, BC BF BF BF 3BF and FC FG FC BF BF 2BF. BC AC 2AD 5 3 DC 5 3AD FC 5 3BF 2 In ABC... |
Complete the proof of Theorem 12.5 (SSS) by showing that DE AB. 17. Prove Theorem 12.6, “Two triangles are similar if the ratios of two pairs of corresponding sides are equal and the corresponding angles included between these sides are congruent. (SAS)” 18. Triangle ABC is an isosceles right triangle with mC 90 and C... |
Similarity A dilation of k is a transformation of the plane such that: 1. The image of point O, the center of dilation, is O. 2. When k is positive and the image of P is P, then h OP and h OPr are the same ray and OP kOP. 3. When k is negative and the image of P is P, then h OP and h OPr are oppo- site rays and OP kOP... |
c 2 ka AC ArCr. Therefore, Therefore, Therefore, Dilations 497 Slope of BC 5 d 2 e b 2 c Slope of BrCr 5 kd 2 ke kb 2 kc BC BrCr. We have shown that AB ArBr and AC ArCr. Therefore, because they are corresponding angles of parallel lines: mOAB mOAB mOAC mOAC mOAB mOAC mOAB mOAC mBAC mBAC In a similar way we can prove th... |
d kp 2 kb is true, Since Thus, since we have shown that the slope of P is on and collinearity is preserved ArBr k k A or kc 2 kq ka 2 kp 5 ArPr kd 2 kq kb 2 kp is equal to the slope of PrBr, EXAMPLE 1 The coordinates of parallelogram EFGH are E(0, 0), F(3, 0), G(4, 2), and H(1, 2). Under D3, the image of EFGH is EFGH. ... |
, 6) 3x, 1 1 3y A 4. (5, 0) B to find the coordinates of the image of each given point. 5. (18, 3) 6. (1, 7) In 7–10, find the coordinates of the image of each given point under D3. 10. 9. (4, 7) 8. (2, 13) 7. (8, 8) 3, 5 1 8 B A In 11–14, each given point is the image under D2. Find the coordinates of each preimage. 1... |
angle, the angles are congruent. h BC h ED, g BEG h EF h BA, and Given: Prove: ABC DEF H B C D E F G 28. The vertices of rectangle ABCD are A(2, 3), B(4, 3), C(4, 1), and D(2, 1). a. Find the coordinates of the vertices of ABCD, the image of ABCD under D5. b. Show that ABCD is a parallelogram. c. Show that ABCD ABCD. ... |
. ArBr and,. and CD be two nonvertical, B(c, d) parallel segments with endpoints, and, A(a, b) D(c 1 e, d) show that the images CrDr are also parallel. C(a 1 e, b). Under the dilation Dk, ArBr and y O B(a, bd) D(c, bd) A(a, b) C(c, b) y O B(c, d) D(ce, d) A(a, b) C(ae, b) x x 14365C12.pgs 7/10/07 8:56 AM Page 502 502 R... |
ilitude k : 1,M is the midpoint, M is the midpoint of ArCr, of BC a, BC a, BM m, and BM m Prove mr 5 a m ar 5 k 1 Strategy Here we can use SAS to prove BCM BCM. Theorem 12.13 If two triangles are similar, the lengths of corresponding angle bisectors have the same ratio as the lengths of any two corresponding sides. Giv... |
365C12.pgs 7/10/07 8:56 AM Page 505 Proportional Relations Among Segments Related to Triangles 505 Proof g AFB Statements g CGD 1. 2. EAB EDC and EBA ECD 3. ABE DCE g EF'AFB g EG'CGD 4. 5. is an altitude from E in is an altitude from 6. 7. EF ABE. EG E in DCE. DC 5 EF AB EG Reasons 1. Given. 2. If two parallel lines ar... |
corresponding sides of two similar triangles is 6 : 7. What is the ratio of the altitudes of the triangles? 8. Corresponding altitudes of two similar triangles have lengths of 9 millimeters and 6 millimeters. If the length of a median of the larger triangle is 24 millimeters, what is the length of a median of the smal... |
Triangle 507 Given and AM BN intersect at P. are medians of ABC that C Prove AP : MP BP : NP 2 : 1 N P M A B Proof Statements Reasons 1. and AM of ABC. BN are the medians 1. Given. 2. M is the midpoint of N is the midpoint of BC. AC and 2. The median of a triangle is a line segment from a vertex to the midpoint of the... |
). Solution A(3, 6) y N(3, 3) P(1, 2) 1 B(9, 0) M(0, 0) O 1 C(9, 0) x (1) Find the coordinates of the midpoint, M, of BC and of the midpoint, N, of AC : coordinates of M coordinates of N 5 29 1 9 2 A 5 (0, 0), 0 1 0 2 5 23 1 9 2 A 5 (3, 3), 6 1 0 2 B B 14365C12.pgs 7/10/07 8:56 AM Page 509 (2) Find the equation of g AM... |
Page 510 510 Ratio, Proportion, and Similarity Developing Skills In 3–10, find the coordinates of the centroid of each triangle with the given vertices. 3. A(3, 0), B(1, 0), C(1, 6) 5. A(3, 3), B(3, 3), C(3, 9) 7. A(1, 1), B(3, 1), C(1, 7) 9. A(2, 5), B(0, 1), C(10, 1) 4. A(5, 1), B(1, 1), C(1, 5) 6. A(1, 2), B(7, 0),... |
projection of g PQ is P. If PR'AB g PQ g. PQ on, the pro- The projection of R on jection of PR on g PQ is P. R A P M B N Q Proportions in the Right Triangle C A D B AB CD'AB AD In the figure, ABC is a right triangle, with the right angle at C. Altitude is so that two smaller triangles are formed, ACD and drawn to hypo... |
5 BC AB. BD and since ABC CBD, C A D B Corollary 12.16b The length of the altitude to the hypotenuse of a right triangle is the mean proportional between the lengths of the projections of the legs on the hypotenuse. Proof: The lengths of the corresponding sides of similar triangles are in proportion. Therefore, since ... |
C12.pgs 7/10/07 8:56 AM Page 514 514 Ratio, Proportion, and Similarity Exercises Writing About Mathematics 1. When altitude is drawn to the hypotenuse of right triangle ABC, it is possible that ACD and BCD are congruent as well as similar. Explain when ACD BCD. CD 2. The altitude to the hypotenuse of right RST separate... |
What are the lengths of the legs of the triangle? 19. In a right triangle whose hypotenuse measures 50 centimeters, the shorter leg measures 30 centimeters. Find the measure of the projection of the shorter leg on the hypotenuse. 20. The segments formed by the altitude to the hypotenuse of right triangle ABC measure 8... |
365C12.pgs 7/10/07 8:56 AM Page 516 516 Ratio, Proportion, and Similarity The Converse of the Pythagorean Theorem If we know the lengths of the three sides of a triangle, we can determine whether the triangle is a right triangle by using the converse of the Pythagorean Theorem. Theorem 12.17b If the square of the lengt... |
celes triangle. The length of the hypotenuse of the triangle is the slant height of the cone, the length one of the legs is the height of the cone, and the length of the other leg is the radius of the base of the cone. If a cone has a height of 24 centimeters and the radius of the base is 10 centimeters, what is the sl... |
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