Write an equation that can be used to answer the question. Then solve. Round to the nearest tenth if necessary. 1. How far up the tree is the cat?

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1 Write an equation that can be used to answer the question. Then solve. Round to the nearest tenth if necessary. 1. How far up the tree is the cat? Notice that the distance from the bottom of the ladder to the tree, the height of the tree where the cat is, and the ladder form a right triangle. Let h represent the height in the tree where the cat is. Use the Pythagorean Theorem. Replace a with 5, c with 12, and use h for b. Simplify. The equation has two solutions, approximately ±10.9. However, the height must be positive. So, the cat is about 10.9 feet up in the tree. 2. How deep is the water? Notice that the triangle under the boat is a right triangle. Let x represent the depth of the water. Use the Pythagorean Theorem. Replace a with 6, c with 15, and use x for b. Simplify. The equation has two solutions, approximately ±13.7. However, the depth must be positive. So, the water is about 13.7 feet deep. esolutions Manual - Powered by Cognero Page 1

2 Find the missing measure in the figure. Round to the nearest tenth if necessary. 3. The height of the pyramid, the slant height, and half the base form a right triangle. Use the Pythagorean Theorem to find the missing measure. 4. Since length cannot be negative, the slant height of the pyramid is about 11.7 centimeters. The height of the cone, the slant height, and the radius of the base form a right triangle. Use the Pythagorean Theorem to find the missing measure. Since length cannot be negative, the slant height of the pyramid is 15 feet. esolutions Manual - Powered by Cognero Page 2

3 5. Refer to the map of the Woodlands Camp below. Round your answers to the nearest tenth. a. How far is it from Sycamore cabin to Oak cabin? b. A camper in Hickory cabin wants to visit a friend in Elm cabin. How much farther is it if she walks to the Mess Hall first? a. Notice that the triangle formed between Sycamore cabin, Oak cabin, and the Mess Hall is a right triangle. Let d represent the distance from Sycamore cabin to Oak cabin. Use the Pythagorean Theorem. Replace a with 30, c with 50, and use d for b. Simplify. The equation has two solutions, ±40. However, the distance from Sycamore cabin to Oak cabin must be positive. So, Sycamore cabin is 40 yards from Oak cabin. b. Notice that the triangle formed between Hickory cabin, the Mess Hall, and Elm cabin is a right triangle. Let h represent the distance from Hickory cabin to the Mess Hall. Use the Pythagorean Theorem. Replace b with 40, c with 60, and use x for a. Simplify. The equation has two solutions, approximately ±44.7. However, the distance from Hickory cabin to the Mess Hall must be positive. So, Hickory cabin is 44.7 yards from the Mess Hall. If a camper walks from Hickory cabin to the Mess Hall and then to Elm cabin, the camper will walk or 84.7 yards. If the camper walked straight from Hickory cabin to Elm cabin, the walk is only 60 yards. By walking to the Mess Hall first, the camper walks an additional or 24.7 yards. esolutions Manual - Powered by Cognero Page 3

4 6. Justify Conclusions Rodrigo is buying a -foot-long fishing rod for his father for his birthday. He wants to put it in a box so that his dad will not be able to guess what is in the box. The box he wants to use is 4 feet long and 4 feet wide. Will the pole fit in the box? Justify your reasoning. yes; Sample answer: The corner of the box is a right angle. Find the length of the diagonal using the Pythagorean Theorem = Since the fishing rod is 5.5 feet long, it will fit diagonally in the box. 7. Identify Structure How do you use the Pythagorean Theorem? Students should say something about substituting the known values into the Pythagorean Theorem and solving for the unknown. esolutions Manual - Powered by Cognero Page 4

5 8. Model with Mathematics Write a problem that can be solved by using the Pythagorean Theorem. Then explain how to solve the problem. Sample answer: Sam leaves his house. He walks 2 miles north, and then turns and walks 3 miles west. How far is Sam from his house? Use the Pythagorean Theorem. Replace a with 2 and b with 3. Simplify. Sam is about 3.6 miles from his house. esolutions Manual - Powered by Cognero Page 5

6 9. Which One Doesn t Belong? Each set of numbers represents the side measures of a triangle. Identify the set that does not belong with the other three. Explain your reasoning. You know that forms a Pythagorean Triple. Check to see whether the other side measures also represent Pythagorean Triples does not belong because the values do not represent a Pythagorean Triple esolutions Manual - Powered by Cognero Page 6

7 10. Persevere with Problems Suppose a ladder 20 feet long is placed against a vertical wall 20 feet high. How far would the top of the ladder move down the wall by pulling out the bottom of the ladder 5 feet? Explain your reasoning. Look at the diagram. You are trying to find a, the distance from the top of the wall to the top of the ladder after the bottom of the ladder is moved 5 feet from the base of the wall. To find a, first you need to find the distance from the top of the ladder to the base of the wall. The ladder, the wall, and the distance from the bottom of the ladder to the wall form a right triangle. Let x represent the distance from the top of the ladder to the base of the wall. Subtract x from 20 to find a = 0.6 So, if the bottom of the ladder were moved 5 feet away from the wall, the top of the ladder would move down 0.6 feet. 11. Model with Mathematics Write and solve a real-world problem that involves using the Pythagorean Theorem or its converse. See students work. esolutions Manual - Powered by Cognero Page 7

8 12. Write an equation to find how far the bird is from the boy. Then solve the equation. Round to the nearest tenth. Notice that the triangle formed between the bird, the boy s eye, and a point directly beneath the bird is a right triangle. Let x represent the distance of the bird from the boy s eye. Use the Pythagorean Theorem. Replace a with 70, b with 20, and use x for c. Simplify. The equation has two solutions, approximately ±72.8. However, the distance from the bird to the boy s eye must be positive. So, the bird is about 72.8 feet away. 13. A party hat is in the shape of a cone with dimensions shown. Find the height of the hat. Round to the nearest tenth. The height of the cone, the slant height, and the radius of the base form a right triangle. Use the Pythagorean Theorem to find the missing measure. Since length cannot be negative, the height of the hat is about 9.0 inches. esolutions Manual - Powered by Cognero Page 8

9 14. Larry wants to go from his house to his grandmother s house. How much distance is saved if he takes Main Street instead of Market and Exchange? Notice that the block formed by Main Street, Market Street, and Exchange Street is a right triangle. Let d represent the distance along Exchange Street. Use the Pythagorean Theorem. Replace a with 3, c with 5, and use d for b. Simplify. The equation has two solutions, ±4. However, the distance along Exchange Street must be positive. So, the distance from Market Street to Larry s house along Exchange Street is 4 blocks, and the distance to his grandmother's house is or 5 blocks. If Larry went to his grandmother s house by taking Exchange Street and then Market Street, he would walk a distance of or 7 blocks. Larry would save 7 5 or 2 blocks if he went to his grandmother s house using Main Street instead of Market and Exchange. esolutions Manual - Powered by Cognero Page 9

10 15. Suppose Greenville, Rock Hill, and Columbia form a right triangle. What is the distance from Columbia to Greenville? Notice that the distances from Greenville to Rock Hill, Rock Hill to Columbia, and Columbia to Greenville form a right triangle. Let d represent the distance from Columbia to Greenville. Use the Pythagorean Theorem. Replace a with 80, b with 68, and use d for c. Simplify. The equation has two solutions, approximately ± However, the distance from Columbia to Greenville must be positive. So, it is about 105 miles from Columbia to Greenville. esolutions Manual - Powered by Cognero Page 10

11 Persevere with Problems Find the missing measure in the figure. Round to the nearest tenth if necessary. 16. In order to find the diagonal of the prism, you first need to find the diagonal of the base. Label that measurement y and use the Pythagorean Theorem to find y. Let a = 5, b = 12, and c = y. So, the diagonal of the base of the prism is 13 millimeters. Use Pythagorean Theorem and the height of the prism, 5 millimeters, to find the length of the diagonal of the prism. Let a = 5, b = 13, and c = c. So, the missing measure in the prism is 13.9 millimeters. esolutions Manual - Powered by Cognero Page 11

12 17. In order to find the diagonal of the prism, you first need to find the diagonal of the base. Label that measurement y and use the Pythagorean Theorem to find y. Let a = 10, b = 10, and c = y. So, the diagonal of the base of the prism is 14.1 inches. Use Pythagorean Theorem and the height of the prism, 15 inches, to find the length of the diagonal of the prism. Let a = 15, b = 14.1, and c = c. So, the missing measure in the prism is 20.6 inches. 18. Shanise designed a stained glass window in the shape of a kite. Select the correct measures to label the dimensions of the window. esolutions Manual - Powered by Cognero Page 12

13 What is the perimeter of the window? To find the missing dimensions first find the leg of one of the congruent triangles that form the bottom of the stained glass window. Use the Pythagorean Theorem to find the missing side, b. Replace a with 27 and c with 45. Simplify. Next, find the hypotenuse of one of the congruent triangles that form the top of the stained glass window. Use the Pythagorean Theorem to find the hypotenuse, c. Replace a with 15 and b with 36. Simplify. esolutions Manual - Powered by Cognero Page 13

14 Use the fact that some side are congruent to write the remaining three measurements. The top two sides of the stained glass window measure 39 inches and the bottom two sides measure 45 inches. So, the perimeter of the stained glass window is or 168 inches. 19. Brayden is building the model bridge shown. How long must he cut the piece of wood for one of the vertical support beams, represented by x? Notice that the sections of the bridge form right triangles. Let x represent the length of the vertical support beam that Brayden needs to cut. Use the Pythagorean Theorem. Replace a with 6, c with 6.5, and use x for b. Simplify. The equation has two solutions, ±2.5. However, the length of the vertical support beam must be positive. So, Brayden needs to cut a piece of wood that is 2.5 inches for the vertical support beam. esolutions Manual - Powered by Cognero Page 14

15 20. Determine whether a triangle with sides 20 inches, 48 inches, and 52 inches long is a right triangle. Justify your answer. Use the Pythagorean Theorem to determine if the side lengths are in the proper relationship to each other. Replace a with 20, b with 48, and c with Since the equation is true, the sides form a right triangle = 52 2 Estimate to the nearest whole number. Justify your reasoning. The largest perfect square less than 39 is 36. The smallest perfect square greater than 39 is 49. So, is between 6 and 7. Since is closer to than, the best integer estimate for is The largest perfect square less than 146 is 144. The smallest perfect square greater than 146 is 169. So, is between 12 and 13, so is between 12 and 13. Since is closer to than, the best integer estimate for is 12. esolutions Manual - Powered by Cognero Page 15

16 23. The largest perfect cube less than 30 is 27. The smallest perfect cube greater than 30 is 64. So, is between 3 and 4. Since is closer to than, the best integer estimate for is 3. esolutions Manual - Powered by Cognero Page 16