1-3 Distance and Midpoints. ALGEBRA Suppose M is the midpoint of value. 53. FM = 3x 4, MG = 5x 26, FG =? SOLUTION: If M is the midpoint, then FM = MG.
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1 ALGEBRA Suppose M is the midpoint of value. 53. FM = 3x 4, MG = 5x 26, FG =? If M is the midpoint, then FM = MG. Use the given information to find the missing measure or Then, x = 11. FM = 3x 4 = 3(11) 4 = 29 MG = 5x 26 = 5(11) 26 = 29 FG = FM + MG = = 58 esolutions Manual - Powered by Cognero Page 1
2 54. FM = 5y + 13, MG = 5 3y, FG =? If M is the midpoint, then FM = MG. Then y = 1. FM = 5y + 13 = 5( 1) + 13 = 8 MG = 5 3y = 5 3( 1) = 8 FG = FM + MG = = 16 esolutions Manual - Powered by Cognero Page 2
3 55. MG = 7x 15, FG = 33, x =? If M is the midpoint, then. Substitute. Thus MG = Find x, esolutions Manual - Powered by Cognero Page 3
4 56. FM = 8a + 1, FG = 42, a =? If M is the midpoint, then Substitute. So, FM = 21. esolutions Manual - Powered by Cognero Page 4
5 57. BASKETBALL The dimensions of a basketball court are shown below. Suppose a player throws the ball from a corner to a teammate standing at the center of the court. a. If center court is located at the origin, find the ordered pair that represents the location of the player in the bottom right corner. b. Find the distance that the ball travels. a. The center court is located at the origin. Since the court is 94 feet long, each end line is (94) or 47 feet from center court. So, the right corners will have x-coordinate values of 47. Since the court is 50 feet wide, each side line be (50) or 25 feet from center court. So, the bottom corners will have y-coordinate values of 25. Therefore, the coordinates of the bottom right corner are (47, 25). b. Use the Distance Formula to find the distance between (0, 0) and (47, 25). The distance between the center of the court and the bottom right corner is about 53.2 ft. The ball will travel about 53.2 ft. esolutions Manual - Powered by Cognero Page 5
6 CCSS TOOLS Spreadsheets can be used to perform calculations quickly. The spreadsheet below can be used to calculate the distance between two points. Values are used in formulas by using a specific cell name. The value of x 1 is used in a formula using its cell name, A2. Write a formula for the indicated cell that could be used to calculate the indicated value using the coordinates ( x 1, y 1 ) and ( x 2, y 2 ) as the endpoint of a segment. 58. E2; the x-value of the midpoint of the segment To find the midpoint of the segment, use the AVERAGE function. The AVERAGE function sums the specified cells and divides by the number of cells. We want to sum A2 and C2 and divide by two. =AVERAGE(A2, C2) 59. F2; the y-value of the midpoint of the segment To find the midpoint of the segment, use the AVERAGE function. The AVERAGE function sums the specified cells and divides by the number of cells. We want to sum B2 and D2 and divide by two. =AVERAGE(B2, D2) 60. G2; the length of the segment To find the distance of the segment, you need to use the distance formula. The distance formula is not a built in function on the spreadsheet. Remember that (x 2, y 2 ) are stored in (C2, D2) and (x 1, y 1 ) are stored in (A2, B2). Use the SQRT function for the square root. Use the ^ key to raise to a power of 2. You will need to have several sets parenthesis. =SQRT((C2 A2)^2 + (D2 B2)^2) esolutions Manual - Powered by Cognero Page 6
7 Name the point(s) that satisfy the given condition. 61. two points on the x-axis that are 10 units from (1, 8) The y-coordinate of the point on the x-axis is 0. So, the point would be of the form (x, 0). Use the Distance Formula to find an expression for the distance between the points (x, 0) and (1, 8) and equate it to 10. There are two possible values for x, 5 and 7. So, the two points are ( 5, 0) and (7, 0). esolutions Manual - Powered by Cognero Page 7
8 62. two points on the y-axis that are 25 units from ( 24, 3) The x-coordinate of the point on the y-axis is 0. So, the point would be of the form (0, y). Use the Distance Formula to find an expression for the distance between the points (0, y) and ( 24, 3) and equate it to 25. There are two possible values for y, 4 and 10. So, the two points are (0, 4) and (0, 10). esolutions Manual - Powered by Cognero Page 8
9 63. COORDINATE GEOMETRY Find the coordinates of B if B is the midpoint of and C is the midpoint of Use the Midpoint Formula to find the coordinates of C. Use the Midpoint Formula to find the coordinates of B. esolutions Manual - Powered by Cognero Page 9
10 ALGEBRA Determine the value(s) of n. 64. J(n, n+2), K(3n, n 1), JK = 5 Use the Distance Formula to find an expression for JK and equate it to 5. esolutions Manual - Powered by Cognero Page 10
11 65. P(3n, n 7), Q(4n, n + 5), PQ = 13 Use the Distance Formula to find an expression for PQ and equate it to WRITING IN MATH Explain how the Pythagorean Theorem and the Distance Formula are related. Sample answer: The Pythagorean Theorem relates the lengths of the legs of a right triangle to the length of the hypotenuse using the formula c 2 = a 2 + b 2. If you take the square root of the formula, you get Think of the hypotenuse of the triangle as the distance between the two points, the a value as the horizontal distance x 2 x 1, and the b value as the vertical distance y 2 y 1. If you substitute, the Pythagorean Theorem becomes the Distance Formula, esolutions Manual - Powered by Cognero Page 11
12 69. REASONING Is the point one third of the way from ( x 1, y 1 ) to ( x 2, y 2 ) sometimes, always, or never the point? Explain. Sample answer: Choose some points that lie on horizontal, vertical, and diagonal line segments. Use the distance for distance between the first pair of points and the first point and the new point. (x 1, y 1 ) (x 2, y 2 ) Distance between first pair of points Distance between first p ( 3, 0) (6, 0) (0, 1) (0, 13) (0, 0) (6, 0) (9,12) (0, 0) (0, 0) (12, 9) ( 4, 5) (5, 7) (3, 2) (3, 4) Test each pair of distances. Only 2 = and 5 =. So when (x 1, y 1 ) = (0, 0), the point way from (x 1, y 1 ) to (x 2, y 2 ). Therefore, the correct answer is sometimes. Solve each equation. 82. Isolate x. esolutions Manual - Powered by Cognero Page 12
13 Isolate a. 85. Isolate k. esolutions Manual - Powered by Cognero Page 13
14 86. Isolate z. 87. Isolate n. esolutions Manual - Powered by Cognero Page 14
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