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interior. Construct segments from this point to each vertex, forming four triangles. Measure the area of each triangle. Move the point to find a location where all four triangles have equal area. Is there more than one such location? Explain your findings. 20. Explain why x must be 48°. 21. What’s wrong with this pict... |
360 ) r 2 bh Asegment 1_ 2 r R r R R 2 r 2 Aannulus LESSON 8.6 Any Way You Slice It 437 EXAMPLE A Find the area of the shaded sector. 45° 20 cm Solution Asector, or 1 ° 5 4 The sector is, of the circle. 8 6 ° 0 3 • r2 a 60 3 • (20)2 5 4 0 6 3 • 400 1 8 50 The area is 50 cm2. The area formula for a sector. Substitute r... |
radius of the small circle is 8 cm. Find x. 18 r x° x° 13. Suppose the pizza slice in the photo at the beginning of this lesson is a sector with a 36° arc, and the pizza has a radius of 20 ft. If one can of tomato sauce will cover 3 ft2 of pizza, how many cans would you need to cover this slice? LESSON 8.6 Any Way You... |
. 20. If the measure of each exterior angle of a regular polygon is 24°, then the polygon has 15 sides. 21. If the diagonals of a parallelogram bisect its angles, then the parallelogram is a square. 22. If two sides of a triangle measure 25 cm and 30 cm, then the third side must be greater than 5 cm but less than 55 cm... |
of the shapes. Shape Vertices Trapezoid (1, 12), (8, 12), (7, 9), (4, 9) Pentagon (3, 1), (4, 4), (6, 4), (9, 2), (7, 0) Square Triangle (0, 6), (3, 9), (6, 6), (3, 3) (11, 0), (7, 4), (11, 12) You will need ● graph paper ● a ruler ● a penny ● a dime 442 CHAPTER 8 Area Step 1 Step 2 Step 3 Step 4 For each shape, calcu... |
devise a game in which the customer flips a coin onto a red-and-white checkered tablecloth with 1-inch squares. If it lands completely within a square, the customer wins, and doesn’t have to pay the bill. If it lands touching or crossing the boundary of a square, the customer loses. I N I N LIBERTY LIBERTY 1996 1996 S... |
die had odd numbers and the other had even numbers? What if you used 8-sided dice? What if you rolled three 6-sided dice instead of two? Choose one of these scenarios or one that you find interesting to investigate. Make your dice and roll them 20 times. Predict what the graph will look like if you roll the dice 100 t... |
. Calculate the area of each face. If some faces are identical, you only need to find the area of one. 3. Find the total area of all the faces. EXAMPLE A Find the surface area of the rectangular prism. 2 m 4 m 5 m Solution First, draw and label all six faces. Then, find the areas of all the rectangular faces. These shi... |
2 and 3 to write a formula for the surface area of a regular n-gon pyramid in terms of base length b, slant height l, and apothem a. Write another expression for the surface area of a regular n-gon pyramid in terms of height l, apothem a, and perimeter of the base, P 1_ 2 P You can find the surface area of a cone usin... |
area of this obelisk. 450 CHAPTER 8 Area 12. APPLICATION Claudette and Marie are planning to paint the exterior walls of their country farmhouse (all vertical surfaces) and to put new cedar shingles on the roof. The paint they like best costs $25 per gallon and covers 250 square feet per gallon. The wood shingles cost... |
you have found a solution, use a diagram to explain it 452 CHAPTER 8 Area Alternative Area Formulas In ancient Egypt, when the yearly floods of the Nile River receded, the river often followed a different course, so the shape of farmers’ fields along the banks could change from year to year. Officials then needed to m... |
all triangles? Step 9 Devise a way of calculating the area of any quadrilateral. Use Sketchpad to test your method. IMPROVING YOUR VISUAL THINKING SKILLS Cover the Square Trace each diagram below onto another sheet of paper. Cut out the four triangles in each of the two small equal squares and arrange them to exactly ... |
each term. 11. Apothem 12. Annulus 13. Sector of a circle For Exercises 14–16, draw a diagram and explain in a paragraph how you derived the area formula for each figure. 14. Parallelogram 15. Trapezoid 16. Circle CHAPTER 8 REVIEW 455 EW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHA... |
nearest tenth of a square centimeter if the apothem measures 6.9 cm and each side measures 10 cm. 36. Find three noncongruent polygons, each with an area of 24 square units, on a 6-by-6 geoboard or a 6-by-6 square dot grid. 37. Lancelot wants to make a pen for his pet, Isosceles. What is the area of the largest rectan... |
to the nearest square foot? For Exercises 44–46, unless the dimensions indicate otherwise, assume each quadrilateral is a rectangle. 44. You are producing 10,000 of these metal wedges, and you must electroplate them with a thin layer of high-conducting silver. The measurements shown are in centimeters. Find the total ... |
perimeter of the base. 6. Use algebra to show that the total surface area of a cylinder is given by the formula SA C(h r), where h is the height of the cylinder, r is the radius of the base, and C is the circumference of the base. 7. Here is a different formula for the area of a trapezoid: A mh, where m is the length ... |
. C. Escher, 1961 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● discover the Pythagorean Theorem, one of the most important concepts in mathematics ● use the Pythagorean Theorem to calculate the distance between any two points ● use conjectures related to the Pythagorean Theorem t... |
In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. If a and b are the lengths of the legs, and c is the length of the hypotenuse, then?. History Pythagoras of Samos (ca. 569–475 B.C.E.), depicted in this statue, is often described as “the first pur... |
acute angles in the right triangle, along with any angle of the quadrilateral, add up to 180°. The acute angles in a right triangle add up to 90°. Therefore the quadrilateral angle measures 90° and the quadrilateral is a square. If it is a square with side length c, then its area is c2. So, a2 b2 c2, which proves the ... |
. s? 6. c? s s 24 10 6 6 8 a c 9. The base is a circle. x? 41 x 9 7. b? 8. x? 8 7 25 b 3.9 1.5 x 1.5 10. s? 11. r? 5 s y r (5, 12) (0, 0) x 12. A baseball infield is a square, each side measuring 90 feet. To the nearest foot, what is the distance from home plate to second base? Second base 13. The diagonal of a square ... |
previous one. Flip books are basic to animation technique. For more information about flip books, see www.keymath.com/DG. These five frames start off the photo series titled The Horse in Motion, by photographer, innovator, and motion picture pioneer Eadweard Muybridge (1830–1904). Here are two dissections that you can... |
triples from the list above. Mark off four points, A, B, C, and D, on a string to create three consecutive lengths from your set of triples. 8 cm B A 15 cm 17 cm D 1 2 3 4 6 7 8 9 11 12 13 14 16 17 15 10 5 C 0 Step 2 Step 3 Loop three paper clips onto the string. Tie the ends together so that points A and D meet. Thre... |
b2 c2 Show: ABC is a right triangle Plan: Begin by constructing a second triangle, right triangle DEF (with F a right angle), with legs of lengths a and b and hypotenuse of length x. The plan is to show that x c, so that the triangles are congruent. Then show that C and F are congruent. Once you show that C is a right... |
the plan to complete the proof. 20 cm 40 cm 60 cm 19. What’s wrong with this picture? 20. Explain why ABC is a right triangle. 3.75 cm 2 cm 4.25 cm Review 21. Identify the point of concurrency from the construction marks. LESSON 9.2 The Converse of the Pythagorean Theorem 471 22. Line CF is tangent to circle D at C. T... |
of prime factors. Look for any square factors (factors that appear twice). 50 5 5 2 52 2 52 2 5 2 Squaring and taking the square root are inverse operations— they undo each other. So, equals 5. 52 You might argue that 52 doesn’t look any simpler than 50. However, in the days before calculators with square root buttons... |
equal to the square root of the third side. THE SCARECROW IN THE 1939 FILM THE WIZARD OF OZ Two Special Right Triangles In this lesson you will use the Pythagorean Theorem to discover some relationships between the sides of two special right triangles. One of these special triangles is an isosceles right triangle, als... |
mB? What are mACD and mBCD? What are mADC and mBDC? Is ADC BDC? Why? Is AD BD? Why? How do AC and AD compare? In a 30°-60°-90° triangle, will this relationship between the hypotenuse and the shorter leg always hold true? Explain. D A B Sketch a 30°-60°-90° triangle. Choose any integer for the length of the shorter leg... |
angles 477 7. The solid is a cube. d? H d F D 12 cm B E A G C 8. g?, h? 9. What is the area of the triangle? 30° h g 120 130 10. Find the coordinates of P. 11. What’s wrong with this picture? y P (?,?) 45° x (1, 0) (0, –1) 8 60° 15 30° 17 12. Sketch and label a figure to demonstrate that 27 is equivalent to 33. (Use is... |
of the sides as centers. Find the area of each semicircle. What relationship do you notice among the three areas? 21. The Jiuzhang suanshu is an ancient Chinese mathematics text of 246 problems. Some solutions use the gou gu, the Chinese name for what we call the Pythagorean Theorem. The gou gu reads gou2 gu2 (xian)2.... |
same angle measures. Why? Though different in size, the three triangles all have the same shape. Figures that have the same shape but not necessarily the same size are called similar figures. You’ll use these similar triangles to prove the Pythagorean Theorem in a later chapter. a b c a b A beautifully complex fractal... |
do you add between Stage 1 and Stage 2? c. How much area is added at any new stage? d. A true fractal exists only after an infinite number of stages. If you could build a true fractal based on the construction in this activity, what would be its total area? Step 5 Give the same color and shade to sets of squares that ... |
in. by 18 in. Will the box be big enough? 482 CHAPTER 9 The Pythagorean Theorem 3. Meteorologist Paul Windward and geologist Rhaina Stone are rushing to a paleontology conference in Pecos Gulch. Paul lifts off in his balloon at noon from Lost Wages, heading east for Pecos Gulch Conference Center. With the wind blowing... |
them straight up. Work is a measure of force applied over distance, and you calculate it as a product of force and distance. For example, a force of 100 pounds is required to hold up a 100-pound object. The work required to lift it 2 feet is 200 foot-pounds. But if you use a 4-foot-long ramp to roll it up, you’ll do t... |
x 45° A C 36 cm 30° B 15. The two rays are tangent to the circle. What’s wrong with this picture? A B 54° C D 226° 16. In the figure below, point A is the image of point A after a reflection over OT. What are the coordinates of A? 17. Which congruence shortcut can you use to show that ABP DCP? 18. Identify the point o... |
graph paper Investigation 1 The Distance Formula In Steps 1 and 2, find the length of each segment by using the segment as the hypotenuse of a right triangle. Simply count the squares on the horizontal and vertical legs, then use the Pythagorean Theorem to find the length of the hypotenuse. Step 1 Copy graphs a–d from... |
289 AB 17 The distance formula. Substitute 8 for x1, 15 for y1, 7 for x2, and 23 for y2. Subtract. Square 15 and 8 and add. Take the square root of both sides. The distance formula is also used to write the equation of a circle. EXAMPLE B Write an equation for the circle with center (5, 4) and radius 7 units. Solution... |
, 16) 2. (15, 37), (42, 73) 3. (19, 16), (3, 14) 4. Look back at the diagram of Viki’s and Scott’s locations on page 486. Assume each block is approximately 50 meters long. What is the shortest distance from Viki to Scott to the nearest meter? 5. Find the perimeter of ABC with vertices A(2, 4), B(8, 12), and C(24, 0). ... |
’s surface. Antonio pulls the lily to one side, keeping the stem straight, until the blossom touches the water at a spot 40 cm from where the stem first broke the water’s surface. How is Antonio able to calculate the depth of the water? What is the depth? 17. CURT is the image of CURT under a R' rotation transformation... |
. Trace along the graph, starting at x 0. Record values (rounded to the nearest 0.1 unit) for the height reached by the ladder when x 3, 6, 9, and 12. If you move the foot of the ladder away from the wall 3 feet at a time, will each move result in the same change in the height reached by the ladder? Explain. Find the v... |
cm and BC 8 cm, then AC 10 cm. Therefore the radius of the circle is 5 cm and the area of the circle is 25 cm2. is a semicircle and AC is a 492 CHAPTER 9 The Pythagorean Theorem EXERCISES In Exercises 1–4, find the area of the shaded region in each figure. Assume lines that appear tangent are tangent at the labeled po... |
10. AB 6 cm C 11. DE 23 cm F A B D E 12. The Gothic arch is based on the equilateral triangle. If the base of the arch measures 80 cm, what is the area of the shaded region? 13. Each of three circles of radius 6 cm is tangent to the other two, and they are inscribed in a rectangle, as shown. What is the height of the ... |
495 ● CHAPTER 11 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER CHAPTER 9 R E V I E W If 50 years from now you’ve forgotten everything else you learned in geometry, you’ll probably still remember the Pythagorean Theorem. (Though let’s hope you don’t really forget everything else!) That’s bec... |
1 cm2. 14. Determine whether ABC with vertices A(3, 5), B(11, 3), and C(8, 8) is an equilateral, isosceles, or isosceles right triangle 15. Sagebrush Sally leaves camp on her dirt bike traveling east at 60 km/hr with a full tank of gas. After 2 hours, she stops and does a little prospecting—with no luck. So she heads n... |
tangency. r r 35 feet 12 feet 21. APPLICATION Read the Technology Connection above. What is the maximum broadcasting radius from a radio tower 1800 feet tall (approximately 0.34 mile)? The radius of Earth is approximately 3960 miles, and you can assume the ground around the tower is nearly flat. Round your answer to t... |
regular nonagon? a regular 20-gon? What is the general rule? 32. A bug clings to a point two inches from the center of a spinning fan blade. The blade spins around once per second. How fast does the bug travel in inches per second? In Exercises 33–40, identify the statement as true or false. For each false statement, ... |
h C. A 2bh B. A 1 bh 2 D. A b2h 43. If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle must be a(n)? triangle. A. right C. obtuse B. acute D. scalene 44. The ordered pair rule (x, y) → (y, x) is a?. A. reflection over the x-axis C. reflection over the line y x B. reflect... |
to the nearest square foot. 65 cm 25 cm 36 cm x cm TAKE ANOTHER LOOK 1. Use geometry software to demonstrate the Pythagorean Theorem. Does your demonstration still work if you use a shape other than a square—for example, an equilateral triangle or a semicircle? 2. Find Elisha Scott Loomis’s Pythagorean Proposition and... |
awaken only astonishment in my viewers. Sometimes “beauty” is a nasty business. M. C. ESCHER Verblifa tin, M. C. Escher, 1963 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● explore and define many three-dimensional solids ● discover formulas for finding the volumes of prisms, pyra... |
the face by naming the polygon that encloses it. A segment where two faces intersect is called an edge. The point of intersection of three or more edges is called a vertex of the polyhedron. Edge Face Vertex Just as a polygon is classified by its number of sides, a polyhedron is classified by its number of faces. The ... |
Trapezoidal pyramid Hexagonal pyramid Square pyramid Like prisms, pyramids are also classified by their bases. The pyramids of Egypt are square pyramids because they have square bases. The altitude of the pyramid is the perpendicular segment from its vertex to the plane of its base. The length of the altitude is the h... |
shaped like cones. Like a pyramid, a cone has a base and a vertex. The base of a cone is a circle and its interior. The radius of a cone is the radius of the base. The vertex of a cone is the point that is the greatest perpendicular distance from the base. The altitude of a cone is the perpendicular segment from the v... |
equilateral triangular region (use the proper marks to show that the base is equilateral) 24. A hexahedron with two trapezoidal faces 25. A cylinder with a height that is twice the diameter of the base (use x and 2x to indicate the height and the diameter) 26. A right cone with a height that is half the diameter of th... |
510 CHAPTER 10 Volume Review For Exercises 38 and 39, how many cubes measuring 1 cm on each edge will fit into the container? 38. A box measuring 2 cm on each inside edge 39. A box measuring 3 cm by 4 cm by 5 cm on the inside edges 40. What is the maximum number of boxes measuring 1 cm by 1 cm by 2 cm that can fit wit... |
or dried peas 512 CHAPTER 10 Volume e. Build a tetrahedron. f. Build an octahedron. g. Build a nonahedron. h. Build at least two different-shaped decahedrons. i. Build at least two different-shaped dodecahedrons. Step 2 Classify all the different polyhedrons your class built as prisms, pyramids, regular polyhedrons, o... |
fill a car’s gas tank or when you fit last night’s leftovers into a freezer dish, you fill the volume of an empty container. Many occupations also require familiarity with volume. An engineer must calculate the volume and the weight of sections of a bridge to avoid too much stress on any one section. Chemists, biologi... |
example, to find the volume of a right triangular prism, find the area of the triangular base (the number of cubes resting on the base) and multiply it by the height (the number of layers of cubes). So you can extend Conjecture A (the volume of right rectangular prisms) to all right prisms and right cylinders. Step 3 ... |
cylinders, regardless of the shapes of their bases. To calculate the volume of a prism or cylinder, first calculate the area of the base using the formula appropriate to its shape. Then multiply the area of the base by the height of the solid. In oblique prisms and cylinders, the lateral edges are no longer at right a... |
V Right trapezoidal prism b2 H h b Right cylinder r H g. V h. V i. V j. V k. V l. V 7, b 6, b2 h 8, r 3 b 9, b2 h 12, r 6 b 8, b2 h 18, r 8 19, 12, H 20 a. V H 20 b. V H 23 c. V For Exercises 8–9, sketch and label each solid described, then find the volume. 8. An oblique trapezoidal prism. The trapezoidal base has a h... |
cubic foot. How many rectangular swimming pools, each 20 feet by 30 feet by 5 feet, could be filled with 250 million gallons of crude oil? The Great Pyramid of Khufu at Giza, Egypt, was built around 2500 B.C.E. 18. When folded, a 12-by-12-foot section of the AIDS Memorial Quilt requires about 1 cubic foot of storage. ... |
would fill from the bottom up with ice, and we would have an ice planet. What a cold thought! 520 CHAPTER 10 Volume 25. Six points are equally spaced around a circular track with a 20 m radius. Ben runs around the track from one point, past the second, to the third. Al runs straight from the first point to the second,... |
a sense of wonder so indestructible that it would last throughout life. RACHEL CARSON You will need ● container pairs of prisms and pyramids ● container pairs of cylinders and cones ● sand, rice, birdseed, or water Step 1 Step 2 Step 3 Step 4 Step 5 Choose a prism and a pyramid that have congruent bases and the same h... |
H. Multiply. The volume is 1443 cm3 or approximately 249.4 cm3. EXAMPLE B A cone has a base radius of 3 in. and a volume of 24 in.3. Find the height. 3 in. LESSON 10.3 Volume of Pyramids and Cones 523 Solution Start with the volume formula and solve for H. V 1 BH 3 V 1 r2(H) 3 24 1 32(H) 3 24 3H 8 H The height of the ... |
each with a volume of 2304 cm3. 13. Mount Fuji, the active volcano in Honshu, Japan, is 3776 m high and has a slope of approximately 30°. Mount Etna, in Sicily, is 3350 m high and approximately 50 km across the base. If you assume they both can be approximated by cones, which volcano is larger? Mount Fuji is Japan’s h... |
2.8 feet tall, how many barrels are needed to hold 17,000 gallons of oil sludge? Recall that a cubic foot of water is about 7.5 gallons. 21. Find the surface area of each of the following polyhedrons. (See the shapes on page 528.) Give exact answers. a. A regular tetrahedron with an edge of 4 cm b. A regular hexahedro... |
hedrons have intrigued mathematicians for thousands of years. Greek philosophers saw the principles of mathematics and science as the guiding forces of the universe. Plato (429–347 B.C.E.) reasoned that because all objects are three-dimensional, their smallest parts must be in the shape of regular polyhedrons. There ar... |
, each having five pentagon-shaped petals around a center pentagon. Complete the net for half a dodecahedron. a b c Now you know what the nets of the five Platonic solids could look like. Let’s use the nets to construct and assemble models of the five Platonic solids. See the Procedure Note for some tips. 1. To save ti... |
.. for this thing we call “failure” is not the falling down, but the staying down. MARY PICKFORD Volume Problems Volume has applications in science, medicine, engineering, and construction. For example, a chemist needs to accurately measure the volume of reactive substances. A doctor may need to calculate the volume of... |
the triangle’s height from that side is 6 cm. 4. A trapezoidal pyramid has a volume of 3168 cm3, and its height is 36 cm. The lengths of the two bases of the trapezoidal base are 20 cm and 28 cm. What is the height of the trapezoidal base? 5. The volume of a cylinder is 628 cm3. Find the radius of the base if the cyli... |
pool hold? Round your answer to the nearest 0.1 gallon. 8 in. 12. Madeleine’s hot tub has the shape of a regular hexagonal prism. The chart on the hot-tub heater tells how long it takes to warm different amounts of water by 10°F. Help Madeleine determine how long it will take to raise the water temperature from 93°F t... |
a magic hexagram on the front of a grid of 19 hexagons. When Bert’s magic hexagram (like its cousin the magic square) is completed, it will have the same sum in every straight hexagonal row, column, or diagonal (whether it is three, four, or five hexagons long). For example, B 12 10 is the same sum as B 2 5 6 9, which... |
2.81 g/cm3 8.89 g/cm3 8.97 g/cm3 21.40 g/cm3 Platinum Density Density Copper Nickel Metal Metal Gold Lead 19.30 g/cm3 Potassium 0.86 g/cm3 11.30 g/cm3 Silver 10.50 g/cm3 Lithium 0.54 g/cm3 Sodium 0.97 g/cm3 LESSON 10.5 Displacement and Density 535 Solution First, find the volume of displaced water. Then, divide the we... |
into a nonreactive liquid in a square prism whose base measures 10 cm on each edge. If the metal is indeed sodium, how high should the liquid level rise? 7. A square-prism container with a base 5 cm by 5 cm is partially filled with water. You drop a clump of metal that weighs 525 g into the container, and the water le... |
the volume of the slice removed from this right cylinder? Give your answer to the nearest cm3. 36 cm 6 cm 8 in. 60° 11. APPLICATION Ofelia has brought home a new aquarium shaped like the regular hexagonal prism shown at right. She isn’t sure her desk is strong enough to hold it. The aquarium, without water, weighs 48 ... |
Drawing If you have ever put together a toy from detailed instructions, or built a birdhouse from a kit, or seen blueprints for a building under construction, you have seen isometric drawings. Isometric means “having equal measure,” so the edges of a cube drawn isometrically all have the same length. In contrast, reca... |
models and draw their isometric and orthographic views. Practice drawing a cube on isometric dot paper. What is the shape of each visible face? Are they congruent? What should the orthographic views of a cube look like? Stack three cubes to make a two-step “staircase.” Turn the structure so that you look at it the way... |
the fraction of the cylinder’s volume that was filled by two hemispheres. What is the formula for the volume of a sphere? State it as your conjecture. Sphere Volume Conjecture The volume of a sphere with radius r is given by the formula?. C-90 EXAMPLE A As an exercise for her art class, Mona has cast a plaster cube, 1... |
cm. The height of your cone is 12 cm. If you push the ice cream into the cone, will all of it fit? 9. APPLICATION Lickety Split ice cream comes in a cylindrical container with an inside diameter of 6 inches and a height of 10 inches. The company claims to give the customer 25 scoops of ice cream per container, each sc... |
150-foot diameter. They plan to ring the building with parking. How far from the building should the parking lot extend? Round your answer to the nearest foot. Parking Building 20. Plot A, B, C, and D onto graph paper. A is (3, 5). C is the reflection of A over the x-axis. B is the rotation of C 180° around the origin... |
early polygons” is a base for a pyramid, and the radius, r, of the sphere is the height of the pyramid. So the volume, V, of the sphere is the sum of the volumes of all the pyramids. Now get ready for some algebra. A horsefly’s eyes resemble spheres covered by “nearly polygons.” Divide the surface of the sphere into 10... |
the volume and total surface area of each solid. All measurements are in centimeters. 1. 2. 1.8 3. You will need Geometry software for Exercises 20 and 21 12 9 LESSON 10.7 Surface Area of a Sphere 547 4. The shaded circle at right has area 40 cm2. Find the surface area of the sphere. 5. Find the volume of a sphere who... |
s of grain this silo will hold. 11. About 70% of Earth’s surface is covered by water. If the diameter of Earth is about 12,750 km, find the area not covered by water to the nearest 100,000 km2. 548 CHAPTER 10 Volume History From the early 13th century to the late 17th century, the Medici family of Florence, Italy, were... |
.. n... 200... LESSON 10.7 Surface Area of a Sphere 549 20. Technology Use geometry software to construct a segment AB, and its midpoint C. Trace C and B, and drag B around to sketch a shape. Compare the shapes they trace. 21. Technology Use geometry software to construct a circle. Choose a point A on the circle and a ... |
between his fingers. If P then Q. P Q If triangle ABC is isosceles, then the base angles are congruent. Triangle ABC is isosceles. Therefore, its base angles are congruent. P: Triangle ABC is isosceles. Q: Triangle ABC’s base angles are congruent. If P then Q. P Q EXPLORATION Sherlock Holmes and Forms of Valid Reasoni... |
with Thurston. Q: Watson had his checkbook with him. If P then Q. Q P P: AC is the longest side in ABC. Q: B is the largest angle in ABC. If P then Q. Q P Activity It’s Elementary! In this activity you’ll apply what you have learned about Modus Ponens (MP) and Modus Tollens (MT). You’ll also get practice using the sym... |
then its diagonals are congruent. c. If yesterday was Thursday, then there is no school tomorrow. There is no school tomorrow. d. If you don’t use Shining Smile toothpaste, then you won’t be successful. You do not use Shining Smile toothpaste. e. If squiggles are flitz, then ruggles are bodrum. Ruggles are not bodrum.... |
All solids are right (not oblique). All measurements are in centimeters. 26 3. 6. 12 20 6 4 4 554 CHAPTER 10 Volume 21 4. 7. 14 12 10 5. 12 12 10 8. 8 15 6 ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAP For Exercises 9–12, calculate each unknown length given the volume of the so... |
APPLICATION Rosa Avila is a plumbing contractor. She needs to deliver 200 lengths of steel pipe to a construction site. Each cylindrical steel pipe is 160 cm long, has an outer diameter of 6 cm, and has an inner diameter of 5 cm. Rosa needs to know if her quarter-tonne truck can handle the weight of the pipes. To the ... |
.36 pounds. Find the thickness of the ball. 556 CHAPTER 10 Volume ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAP 27. A water barrel that is 1 m in diameter and 1.5 m long is partially filled. By tapping on its sides, you estimate that the water is 0.25 m deep at the deepest point... |
If not, which would be greater, the surface area of the smooth sphere or of the bumpy sphere? Explain. Assessing What You’ve Learned Spaceship Earth, the Epcot center’s signature structure, opened in 1982 in Walt Disney World. It features a ride that chronicles the history of communication technology. UPDATE YOUR PORT... |
BRA SKILLS 9 ● USING Y USING YOUR ALGEBRA SKILLS 9 Proportion and Reasoning Working with similar geometric figures involves ratios and proportions. You may be a little rusty with these topics, so let’s review. A ratio is an expression that compares two quantities by division. You can write the ratio of quantity a to qu... |
Multiply both sides by 20. Multiply and divide on the left side. Subtract 50 from both sides. EXERCISES 1. Look at the rectangle at right. Find the ratio of the shaded area to the area of the whole figure. Find the ratio of the shaded area to the unshaded area. D D, and B C, C A. 2. Use the figure below to find these ... |
, 10 cm, 13 cm, 16 cm, 14 cm, and 12 cm. Zenor is 130 years old, and her seven antennae have an average length of 17 cm. How old is Altor? C B B 18. Assume A. Find AB and BC. Y Z XY 10.5 cm A B C 2 cm Y X 5 cm Z IMPROVING YOUR ALGEBRA SKILLS Algebraic Magic Squares II In this algebraic magic square, the sum of the entr... |
ement of hexagon ABCDEF—they are similar Step 1 Step 2 Step 3 Use patty paper to compare all corresponding angles. How do the corresponding angles compare? Measure the corresponding segments in both hexagons. Find the ratios of the lengths of corresponding sides. How do the ratios of corresponding sides compare? Simila... |
L 18 ft 78° S 83° M E 24 ft B G x y I Solution The quadrilaterals are similar, so you can use a proportion to find x. 1 2 1 8 4 2 x 18x (24)(21) x 28 A proportion of corresponding sides. Multiply both sides by 24x and reduce. Divide both sides by 18. The measure of the side labeled x is 28 ft. In similar polygons, cor... |
of the largest scale drawings ever made. This cave art is part of a grouping of over 15,000 drawings in Tassili N’Ajjier National Park of the Algerian Sahara. Interestingly, these drawings depict animals and landscapes that are absent from the region today, such as these elephants or vast lakes. This monkey is a geogl... |
ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle? 16. Copy this quadrilateral onto your graph paper. Draw a similar quadrilateral with each side half the length of its corresponding side in the original quadrilateral. y Y (2, 5) R (2, 2) O (6, 2) x 17. APPLICATION The photo at rig... |
rotation using a compass and straightedge. Explain your method. Career Similarity plays an important part in the design of cars, trucks, and airplanes, which is done with small-scale drawings and models. This model airplane is about to be tested in a wind tunnel. 570 CHAPTER 11 Similarity MAKING A MURAL A mural is a l... |
, with D A and E B. What will be true about C and F? Why? Carefully measure the lengths of the sides of both triangles. Compare the ratios C B C A AB? of the corresponding sides. Is E F F D D E Compare your results with the results of others near you. You should be ready to state a conjecture. AA Similarity Conjecture ... |
, 30) (15, y) (5, 3) x LESSON 11.2 Similar Triangles 575 Review 16. In the figure below right, find the radius, r, of one of the small circles in terms of the radius, R, of the large circle. R r This Tibetan mandala is a complex design with a square inscribed within a circle and tangent circles inscribed within the cor... |
between 315 and 330 C.E., and broke when sculptors tried to add the extra weight of a beard to its face. The pieces of the statue remain close to its original location in Rome, Italy. IMPROVING YOUR VISUAL THINKING SKILLS Build a Two-Piece Puzzle Construct two copies of Figure A, shown at right. Here’s how to construc... |
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