Find each missing length. If necessary, round to the nearest hundredth.

Transcription

1 Find each missing length. If necessary, round to the nearest hundredth. 1. Use the Pythagorean Theorem, substituting 3 for a and 4 for b.. Use the Pythagorean Theorem, substituting 4 for a and 1 for c. 3. Use the Pythagorean Theorem, substituting 6 for b and 19 for c. Page 1

2 3. Use the Pythagorean Theorem, substituting 6 for b and 19 for c. 4. Use the Pythagorean Theorem, substituting 8 for a and 1 for b. 5.BASEBALL A baseball diamond is a square. The distance between consecutive bases is 90 feet. a. How far does a catcher have to throw the ball from home plate to second base? b. How far does a third baseman throw the ball to the first baseman from a point in the baseline 15 feet from third to second base? c. A base runner going from first to second base is 100 feet from home plate. How far is the runner from second base? a. Use the Pythagorean Theorem, substituting 90 for a and 90 for b. Page

3 a. How far does a catcher have to throw the ball from home plate to second base? b. How far does a third baseman throw the ball to the first baseman from a point in the baseline 15 feet from third second base? 10-5toThe Pythagorean Theorem c. A base runner going from first to second base is 100 feet from home plate. How far is the runner from second base? a. Use the Pythagorean Theorem, substituting 90 for a and 90 for b. The catcher has to throw the ball about 17 ft from home plate to second base. b. The diagram below illustrates the throw made by the third baseman. The length of a is or 75 feet. Find the length of the throw c by using the Pythagorean Theorem, substituting 75 for a and 90 for b. The third baseman has to throw the ball about 117 feet to the first baseman. c. The diagram below illustrates the position of the base runner. Use the Pythagorean Theorem to find the length of side a by substituting 90 for b and 100 for c. Page 3

4 Use the Pythagorean Theorem to find the length of side a by substituting 90 for b and 100 for c. So, the base runner is about 44 feet from first base to second base. Therefore, the distance the base runner is from second is about or 46 feet. Determine whether each set of measures can be the lengths of the sides of a right triangle. 6.8, 1, 16 Since the measure of the longest side is 16, let c = 16, a = 8, and b = 1. Then determine whether c = a + b. No, because c a + b, a triangle with side lengths 8, 1, and 16 is not a right triangle. 7.8, 45, 53 Since the measure of the longest side is 53, let c = 53, a = 8, and b = 45. Then determine whether c = a + b. Yes, because c = a + b, a triangle with side lengths 8, 45, and 53 is a right triangle. 8.7, 4, 5 Since the measure of the longest side is 5, let c = 5, a = 7, and b = 4. Then determine whether c = a + b. Yes, because c = a + b, a triangle with side lengths 7, 4, and 5 is a right triangle. esolutions Manual 9.15, 5, 45 - Powered by Cognero Page 4

5 Yes, because c = a + b, a triangle with side lengths 7, 4, and 5 is a right triangle. 9.15, 5, 45 Since the measure of the longest side is 45, let c = 45, a = 15, and b = 5. Then determine whether c = a + b. No, because c a + b, a triangle with side lengths 15, 5, and 45 is not a right triangle. Find each missing length. If necessary, round to the nearest hundredth. 10. Use the Pythagorean Theorem, substituting 6 for a and 14 for b. 11. Use the Pythagorean Theorem, substituting for a and 1 for c. 1. Page 5

6 1. Use the Pythagorean Theorem, substituting fora and 0 for b. 13. Use the Pythagorean Theorem, substituting 9 for b and 31 for c. 14. Use the Pythagorean Theorem, substituting 16 for a and 6 for c. Page 6

7 14. Use the Pythagorean Theorem, substituting 16 for a and 6 for c. 15. Use the Pythagorean Theorem, substituting fora and forb. 16. Use the Pythagorean Theorem, substituting 7 for b and 5 for c. Page 7

8 16. Use the Pythagorean Theorem, substituting 7 for b and 5 for c. 17. Use the Pythagorean Theorem, substituting 5 for a and forc. 18. Use the Pythagorean Theorem, substituting forb and forc. Page 8

9 18. Use the Pythagorean Theorem, substituting forb and forc. 19.TELEVISION Larry is buying an entertainment stand for his television. The diagonal of his television is 4 inches. The space for the television measures 30 inches by 36 inches. Will Larry s television fit? Explain. Use the Pythagorean Theorem, substituting 30 for a and 36 for b. Yes; sample answer: The diagonal of the space in the TV stand is about 46.9 inches, which is greater than the diagonal of the television, so Larry s TV will fit. Determine whether each set of measures can be the lengths of the sides of a right triangle. Then determine whether they form a Pythagorean triple. 0.9, 40, 41 Since the measure of the longest side is 41, let c = 41, a = 9, and b = 40. Then determine whether c = a + b. Yes, because c = a + b, a triangle with side lengths 9, 40, and 41 is a right triangle. Yes, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 1. Page 9

10 10-5Yes; The sample Pythagorean answer:theorem The diagonal of the space in the TV stand is about 46.9 inches, which is greater than the diagonal of the television, so Larry s TV will fit. Determine whether each set of measures can be the lengths of the sides of a right triangle. Then determine whether they form a Pythagorean triple. 0.9, 40, 41 Since the measure of the longest side is 41, let c = 41, a = 9, and b = 40. Then determine whether c = a + b. Yes, because c = a + b, a triangle with side lengths 9, 40, and 41 is a right triangle. Yes, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 1. Since the measure of the longest side is, let c =, a = 3, and b =. Then determine whether c = a + b. No, because c a + b, a triangle with side lengths 3,, and isnotarighttriangle. No, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number.. Since the measure of the longest side is 1, let c = 1, a = 4, and b = No, because c a + b, a triangle with side lengths 4,. Then determine whether c = a + b., and 1 is not a right triangle. No, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 3. Since the measure of the longest side is 14, let c = 14, a = Page 10, and b = 7. Then determine whether c = a + b.

11 No, because c a + b, a triangle with side lengths 4,, and 1 is not a right triangle. 10-5No, Thebecause Pythagorean Theorem a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 3. Since the measure of the longest side is 14, let c = 14, a = No, because c a + b, a triangle with side lengths, and b = 7. Then determine whether c = a + b., 7, and 14 is not a right triangle. No, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 4.8, 31.5, 3.5 Since the measure of the longest side is 3.5, let c = 3.5, a = 8, and b = Then determine whether c = a + b. Yes, because c = a + b, a triangle with side lengths 8, 31.5, and 3.5 is a right triangle. No, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number, but b and c are not whole numbers. 5. Since the measure of the longest side is, let c =,a =, and b =. Then determine whether c = a +b. No, because c a + b, a triangle with side lengths,, and isnotarighttriangle. No, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 6.18, 4, 30 Since the measure of the longest side is 30, let c = 30, a = 18, and b = 4. Then determine whether c = a + b. Page 11

12 No, because c a + b, a triangle with side lengths,, and isnotarighttriangle. 10-5No, Thebecause Pythagorean Theorem a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 6.18, 4, 30 Since the measure of the longest side is 30, let c = 30, a = 18, and b = 4. Then determine whether c = a + b. Yes, because c = a + b, a triangle with side lengths 18, 4, and 30 is a right triangle. Yes, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 7.36, 77, 85 Since the measure of the longest side is 85, let c = 85, a = 36, and b = 77. Then determine whether c = a + b. Yes, because c = a + b, a triangle with side lengths 36, 77, and 85 is a right triangle. Yes, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 8.17, 33, 98 Since the measure of the longest side is 98, let c = 98, a = 17, and b = 33. The length of c is greater than the sum of a and b,sothisisnotatriangle. No, because c a + b, a triangle with side lengths 17, 33, and 98 is not a triangle. No, because a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 9.GEOMETRY Refer to the triangle shown. a. What is a? b. Find the area of the triangle. a. Use the Pythagorean Theorem, substituting 11 for b and 3 for c. Page 1

13 a and b,sothisisnotatriangle. No, because c a + b, a triangle with side lengths 17, 33, and 98 is not a triangle. 10-5No, Thebecause Pythagorean Theorem a Pythagorean triple is a group of three whole numbers that satisfy the equation c = a + b, where c is the greatest number. 9.GEOMETRY Refer to the triangle shown. a. What is a? b. Find the area of the triangle. a. Use the Pythagorean Theorem, substituting 11 for b and 3 for c. b. The area of the triangle is about units. 30.GARDENING Khaliah wants to plant flowers in a triangular plot. She has three lengths of plastic garden edging that measure 8 feet, 15 feet, and 17 feet. Determine whether these pieces form a right triangle. Explain. Yes; sample answer: The longest side is 17 feet so this has to be the hypotenuse = 17 so the pieces form a right triangle by the converse of the Pythagorean Theorem. 31.LADDER Mr. Takeo is locked out of his house. The only open window is on the second floor. There is a bush along the edge of the house, so he places the neighbor s ladder 10 feet from the house. To the nearest foot, what length of ladder does he need to reach the window? Use the Pythagorean Theorem, substituting 10 for a and 8 for b. Page 13

14 = 17 so the pieces form a right triangle by the converse of the Pythagorean Theorem. 31.LADDER Mr. Takeo is locked out of his house. The only open window is on the second floor. There is a bush along the edge of the house, so he places the neighbor s ladder 10 feet from the house. To the nearest foot, what length of ladder does he need to reach the window? Use the Pythagorean Theorem, substituting 10 for a and 8 for b. He will need a 30-ft ladder to reach the window. CCSSTOOLSFind the length of the hypotenuse. Round to the nearest hundredth. 3. Use the Pythagorean Theorem, substituting 7 for a and 10 for b. 33. Page 14

15 33. Use the Pythagorean Theorem, substituting 4 for a and 7 for b. 34.DOLLHOUSE Alonso is building a dollhouse for his sister s birthday. The roof is 4 inches across and the slanted side is 16 inches long as shown. Find the height of the roof to the nearest tenth of an inch. Use the Pythagorean Theorem, substituting 1 for b and 16 for c. The height of the roof is 10.6 in. 35.GEOMETRY Each side of a cube is 5 inches long. Find the length of a diagonal of the cube. Use the Pythagorean Theorem to find the diagonal of a side of the cube, substituting 5 for a and 5 for b. Page 15

16 The height of the roof is 10.6 in. 35.GEOMETRY Each side of a cube is 5 inches long. Find the length of a diagonal of the cube. Use the Pythagorean Theorem to find the diagonal of a side of the cube, substituting 5 for a and 5 for b. Then, use the Pythagorean Theorem to find a diagonal of the cube, substituting 5 for a and forb. The length of a diagonal of the cube is about 8.66 in. 36.TOWNSQUARES The largest town square in the world is Tiananmen Square in Beijing, China, covering 98 acres. a. One square mile is 640 acres. Assuming that Tiananmen Square is a square, how many feet long is a side to the nearest foot? b. To the nearest foot, what is the diagonal distance across Tiananmen Square? a. First, use the conversion factor 1 square mile = 640 acres to convert the area of the square from acres to square miles. So, 98 acres is square miles. Use dimensional analysis to convert this measure to square feet. Manual - Powered by Cognero esolutions Next, use the formula for the area of a square to find the length of a side. Page 16

17 The length of a diagonal of the cube is about 8.66 in. 36.TOWNSQUARES The largest town square in the world is Tiananmen Square in Beijing, China, covering 98 acres. a. One square mile is 640 acres. Assuming that Tiananmen Square is a square, how many feet long is a side to the nearest foot? b. To the nearest foot, what is the diagonal distance across Tiananmen Square? a. First, use the conversion factor 1 square mile = 640 acres to convert the area of the square from acres to square miles. So, 98 acres is square miles. Use dimensional analysis to convert this measure to square feet. Next, use the formula for the area of a square to find the length of a side. Therefore, a side of Tiananmen Square is about 066 feet long. b. Use the Pythagorean Theorem to find the length of the diagonal of the square, substituting 066 for both a and b. The diagonal distance across Tiananmen Square is about 9 feet. 37.TRUCKS Violeta needs to construct a ramp to roll a cart of moving boxes from her garage into the back of her truck. The truck is 6 feet from the garage. The back of the truck is 36 inches above the ground. How long does the ramp have to be? First, rewrite 36 inches as 3 ft. Use the Pythagorean thereom, substituting 6 for a and 3 for b. The ramp must be about 6.7 ft. Page 17

18 The diagonal distance across Tiananmen Square is about 9 feet. 37.TRUCKS Violeta needs to construct a ramp to roll a cart of moving boxes from her garage into the back of her truck. The truck is 6 feet from the garage. The back of the truck is 36 inches above the ground. How long does the ramp have to be? First, rewrite 36 inches as 3 ft. Use the Pythagorean thereom, substituting 6 for a and 3 for b. The ramp must be about 6.7 ft. If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 38.a = x, b = x + 41, c = 85 Use the Zero Product Property to solve for x. a = 36; b = a = 8, b = x, c = x + b = 15; c = a = 1, b = x, c = x Page 18

19 b = 15; c = a = 1, b = x, c = x b = 35; c = a = x, b = x + 7, c = 97 Use the Zero Product Property to solve for x. a = 65; b = 7 4.a = x 47, b = x, c = x + Use the Zero Product Property to solve for x. a = 16; b = 63; c = a = x 3, b = x 1, c = x Page 19

20 Zero Product Property to solve for x. 10-5Use Thethe Pythagorean Theorem a = 16; b = 63; c = a = x 3, b = x 1, c = x Use the Zero Product Property to solve for x. a = 9; b = 40; c = GEOMETRY A right triangle has one leg that is 8 inches shorter than the other leg. The hypotenuse is 30 inches long. Find the length of each leg. The length of each leg of the triangle is about 4.83 in. and about in. 45.MULTIPLE REPRESENTATIONS In this problem, you will derive a method for finding the midpoint and length of a segment on the coordinate plane. a. Graphical Use a graph to find the lengths of the segments between (3, ) and (8, ) and between (4, 1) and (4, 9). Then find the midpoint of each segment. b. Logical Use what you learned in part a to write expressions for the lengths of the segments between (x1, y) and (x, y) and between (x, y 1 ) and (x,y ). What would be the midpoint of each segment? c. Analytical Based on your results from part b, find the midpoint of the segment with endpoints at (x1, y 1), and (x, y ). d. Analytical Use the Pythagorean Theorem to write an expression for the distance between (x1, y 1), and (x, y ). a.from the graph, the length of the vertical segment is 9 1 or 8 units and the length of the horizontal line is 8 3 or 5 Manual units. - Powered by Cognero esolutions Page 0

21 y ). d. Analytical Use the Pythagorean Theorem to write an expression for the distance between (x1, y 1), and (x, y ). a.from the graph, the length of the vertical segment is 9 1 or 8 units and the length of the horizontal line is 8 3 or 5 units. The midpoints will be located in the middle of each segment, or the average of the coordinates. For the vertical segment, the value between 1 and 9 is 5, so the midpoint should be at (4, 5). For the horizontal segment, the value between 3 and 8 is 5.5, so the midpoint should be at (5.5, ). b.distance: The length of the segment between (x1, y) and (x, y) will be the absolute value of x1 x. The length of the segment between (x, y 1) and (x, y ) will be the absolute value of y 1 y. Midpoint: When the x-coordinates are identical, then the x-coordinate of the midpoint will also be the same value. This is true for the y-coordinates as well. The midpoint of (x1, y) and (x, y) is at the average of the x-coordinates and y or at, and the midpoint of (x, y 1) and (x, y ) is at. c. When the x- and y-coordinates are both different, the midpoint should be at the average values of the coordinates,. or at d.form a right triangle with the segment formed by the two points as the hypotenuse. Note that the other coordinate is (x, y 1)inthedrawing. We know the Pythagorean Theorem is a + b = c. We can solve this formula for c by taking the square root of each side. So,. Now we can replace the variables with the lengths of each side. The length of side a is x x1. The length of side b is y y 1. The length of side c is now. 46.ERRORANALYSIS Wyatt and Dario are determining whether 36, 77, and 85 form a Pythagorean triple. Is either of them correct? Explain your reasoning. Page 1

22 each side. So,. Now we can replace the variables with the lengths of each side. The length of is x x1. TheTheorem length of side b is y y 1. The length of side c is now 10-5side TheaPythagorean. 46.ERRORANALYSIS Wyatt and Dario are determining whether 36, 77, and 85 form a Pythagorean triple. Is either of them correct? Explain your reasoning. Wyatt is correct. The square of the greatest value should be equal to the sum of the squares of the two smaller values.sincethisisthecase,thenumbersformapythagoreantriple. 47.CCSSPERSEVERANCE Find the value of x in the figure shown. The figure can be separated into two right triangles. Let m represent the missing side length common to both right triangles. From the Pythagorean Theorem, m = + x and 14 = 8 + m. Using substitution, you can find that + x = Solve for x. 48.REASONING Provide a counterexample to the statement. Any two right triangles with the same hypotenuse have the same area. Different lengths of a and b can produce the same value of c in the Pythagorean theorem. Page

23 48.REASONING Provide a counterexample to the statement. Any two right triangles with the same hypotenuse have the same area. Different lengths of a and b can produce the same value of c in the Pythagorean theorem. Sample answer: A right triangle with legs measuring 3 cm and 4 cm has a hypotenuse of 5 cm and an area of 6 cm. A right triangle with legs measuring cm and which is not equivalent to 6 cm. cmalsohasahypotenuseof5cm,butitsareais 49.OPENENDED Draw a right triangle that has a hypotenuse of cm, units. Use the Pythagorean Theorem to select two side lengths for a and b with the given value of c. 50.WRITINGINMATH Explain how to determine whether segments in three lengths could form a right triangle. From the converse of the Pythagorean Theorem, if a + b = c then a, b, and c are the lengths of the side of a right triangle. So, check to see whether the square of the greatest number is equal to the sum of the squares of the other two numbers. Considertrianglewithsidesof9,1and15andatrianglewithsidesof7,13and GEOMETRY Find the missing length. A 17 B Page 3

24 51.GEOMETRY Find the missing length. A 17 B C D 17 The correct choice is C. 5.What is a solution of this equation? F 0, 3 G 3 H 0 J no solutions The correct choice is H. 53.SHORTRESPONSE A plumber charges $40 for the first hour of each house call plus $8 for each additional half hour. If the plumber works for 4 hours, how much does he charge? Let x = the total number of hours worked. The plumber charges $88 for 4 hours of work. 54.Find the next term in the geometric sequence. Page 4

25 The plumber charges $88 for 4 hours of work. 54.Find the next term in the geometric sequence. A B C D The correct choice is B. Solve each equation. Check your solution. 55. Check. 56. Check. Page 5

26 Check. 56. Check. 57. Check. 58. Check. 59. Page 6

27 59. Check. 60. Check. There is no real solution. Simplify each expression Page 7

28 Describe how the graph of each function is related to the graph of f (x) = x. Page 8

29 Describe how the graph of each function is related to the graph of f (x) = x. 67.g(x) = x 8 The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of c is 8, and 8 < 0. If c < 0, the graph of f (x) = x is translated parent graph shifted down 8 units. 68.h(x) = x unitsdown.therefore,thegraphof g(x) = x 8 is a translation of the The graph of f (x) = ax stretches or compresses the graph of f (x) = x vertically. The value of a is <1,thegraphoff (x) = x is compressed vertically. Therefore, the graph of h(x) = 1. If 0 < <, and 0 < x is the parent graph compressed vertically. 69.h(x) = x + 5 The graph of f (x) = x reflects the graph of f (x) = x across the x-axis. The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of c is 5, and 5 > 0. If c > 0, the graph of f (x) = x is translated unitsup.therefore,thegraphof h(x) = x + 5 is a translation of the parent graph shifted up 8 units and reflected across the x-axis. 70.g(x) = (x + 10) The graph of f (x) = (x c) represents a horizontal translation of the parent graph. The value of c is 10, and 10 < 0. If c < 0, the graph of f (x) = (x c) is translated unitsleft.therefore,thegraphof g(x) = (x + 10) is a translation of the parent graph shifted left 10 units. 71.g(x) = x Page 9 The graph of f (x) = x reflects the graph of f (x) = x across the x-axis. The graph of f (x) = ax stretches or compresses the graph of f (x) = x vertically. The value of a is, and > 1. If >1,thegraphoff (x) = x is

30 The graph of f (x) = (x c) represents a horizontal translation of the parent graph. The value of c is 10, and 10 < If cpythagorean < 0, the graphtheorem of f (x) = (x c) is translated unitsleft.therefore,thegraphof g(x) = (x + 10) is a The translation of the parent graph shifted left 10 units. 71.g(x) = x The graph of f (x) = x reflects the graph of f (x) = x across the x-axis. The graph of f (x) = ax stretches or compresses the graph of f (x) = x vertically. The value of a is, and > 1. If >1,thegraphoff (x) = x is stretched vertically. Therefore, the graph of g(x) = x is the parent graph reflected across the x-axis and stretched vertically. 7.h(x) = x The graph of f (x) = x reflects the graph of f (x) = x across the x-axis. The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of c is translated unitsdown.therefore,thegraphof h(x) = x, and < 0. If c < 0, the graph of f (x) = x is is a translation of the parent graph shifted down unitsandreflectedacrossthex-axis. 73.ROCKCLIMBING While rock climbing, Damaris launches a grappling hook from a height of 6 feet with an initial upward velocity of 56 feet per second. The hook just misses the stone ledge that she wants to scale. As it falls, the hook anchors on a ledge 30 feet above the ground. How long was the hook in the air? A half second after she throws the hook, it is 30 feet in the air. The hook continues up, then drops back down to the height of 30 feet 3 seconds after the throw. The hook was in the air for 3 seconds. Find each product. 74.(b + 8)(b + ) 75.(x 4)(x 9) 76.(y + 4)(y 8) Page 30

31 75.(x 4)(x 9) 76.(y + 4)(y 8) 77.(p + )(p 10) 78.(w 5)(w + 7) 79.(8d + 3)(5d + ) 80.BUSINESS The amount of money spent at West Outlet Mall continues to increase. The total T(x) in millions of x dollars can be estimated by the function T(x) = 1(1.1), where x is the number of years after it opened in 005. Find the amount of sales in 015, 016, and 017. The sales for the mall in 015 will be about $37.7 million; in 016 will be about $41.74 million; and in 017 will be about $46.75 million. Solve each proportion Page 31

32 10-5The Thesales Pythagorean for the malltheorem in 015 will be about $37.7 million; in 016 will be about $41.74 million; and in 017 will be about $46.75 million. Solve each proportion Page 3