Question

use a problem-solving model to solve each problem.

1. reggie and yoki are riding their bikes to meet at the library. yoki rode 4 miles south and 4 miles west, and reggie rode 8 miles east and 5 miles north as shown below. to the nearest mile, what is the straight-line distance that reggie lives from yoki?

a 6
c 15
b 9
d 225
(image of a coordinate grid with yoki, library, and reggies positions)

1. triangle a has side lengths of 10 units, 24 units, and 26 units. ryan cut out two copies of triangle a and joined them together to form a rectangle. what is the perimeter of the rectangle formed when the two triangles are joined?
2. felix hits three croquet balls from the same spot as shown in the diagram below. starting at point f, he picks up each ball. what is the shortest distance felix can travel to pick up all three croquet balls, ending at the point where he picks up the third ball?

(image of a diagram with ball 1, ball 2, ball 3, and distances 9 yds, 12 yds, 16 yds)

1. what is the perimeter of parallelogram ptru shown in the diagram below? explain.

(image of a diagram with parallelogram ptru, lengths 3 cm, 1 cm, 4 cm)
368 chapter 5 triangles and the pythagorean theorem

Explanation:

Problem 15

Step1: Calculate horizontal distance

Yoki rode 4 miles west (negative x) and Reggie rode 8 miles east (positive x). The horizontal distance between them is \( 8 + 4 = 12 \) miles.

Step2: Calculate vertical distance

Yoki rode 4 miles south (negative y) and Reggie rode 5 miles north (positive y). The vertical distance between them is \( 5 + 4 = 9 \) miles.

Step3: Apply Pythagorean theorem

The straight - line distance \( d \) between them is given by \( d=\sqrt{12^{2}+9^{2}}=\sqrt{144 + 81}=\sqrt{225}=15 \) miles. Wait, no, wait. Wait, Yoki's position: from the origin (library), Yoki is 4 miles east and 4 miles north? Wait, the diagram: Yoki rode 4 miles south and 4 miles west? Wait, no, the diagram shows Yoki is 4 miles east (right) and 4 miles north (up) from the library? Wait, no, the problem says Yoki rode 4 miles south and 4 miles west? Wait, no, the diagram: the library is at O. Yoki is 4 miles east (E) and 4 miles north (N) from O? Wait, the problem statement: "Yoki rode 4 miles south and 4 miles west" – no, the diagram: the arrow for Yoki is 4 miles down (south) and 4 miles right (east)? Wait, I misread. Let's re - read: Yoki rode 4 miles south and 4 miles west? No, the problem says: "Yoki rode 4 miles south and 4 miles west, and Reggie rode 8 miles east and 5 miles north". Wait, the diagram: O is the library. Yoki is at a point: from O, 4 miles east (E) and 4 miles north (N)? No, the arrow for Yoki is 4 miles down (south) and 4 miles right (east)? Wait, the problem statement: "Yoki rode 4 miles south and 4 miles west" – no, maybe the diagram is different. Wait, Reggie rode 8 miles east and 5 miles north? No, Reggie is at a point 8 miles west? Wait, no, the diagram: Reggie is 8 miles west (left) and 5 miles south (down) from O? Wait, I think I made a mistake in horizontal and vertical directions. Let's define the coordinate system with O (library) at (0,0). Let east be positive x, north be positive y. Yoki's movement: 4 miles south (y - 4) and 4 miles west (x - 4), so Yoki's position is (- 4,-4). Reggie's movement: 8 miles east (x + 8) and 5 miles north (y + 5), so Reggie's position is (8,5). Now, the horizontal distance between them is \( 8-(-4)=12 \), vertical distance is \( 5-(-4)=9 \). Then \( d=\sqrt{12^{2}+9^{2}}=\sqrt{144 + 81}=\sqrt{225}=15 \). But the options include 15 (option C). Wait, but earlier I thought I messed up, but now with coordinates: Yoki: x=-4, y = - 4; Reggie: x = 8, y = 5. Difference in x: 8 - (-4)=12, difference in y:5 - (-4)=9. Then distance is \( \sqrt{12^{2}+9^{2}} = 15 \). So the answer is C.

Step1: Identify triangle type

First, check if triangle A is a right triangle. For sides 10, 24, 26: \( 10^{2}+24^{2}=100 + 576 = 676 \), and \( 26^{2}=676 \). So triangle A is a right triangle with legs 10 and 24, hypotenuse 26.

Step2: Determine rectangle dimensions

When two right triangles are joined to form a rectangle, the length and width of the rectangle are equal to the legs of the right triangle. So the length \( l = 24 \) units, width \( w = 10 \) units.

Step3: Calculate perimeter of rectangle

The formula for the perimeter of a rectangle is \( P = 2(l + w) \). Substitute \( l = 24 \) and \( w = 10 \): \( P=2(24 + 10)=2\times34 = 68 \) units.

Step1: Analyze the triangle formed

The positions of the balls form a right triangle. Ball 1 is 9 yds west of F, Ball 3 is 16 yds east of F, so the distance between Ball 1 and Ball 3 is \( 9 + 16=25 \) yds. Ball 2 is 12 yds north of F. So the three balls form a right triangle with legs 12 yds and 25 yds? Wait, no. Wait, starting at F, to pick up Ball 1 (9 yds west), then Ball 2 (12 yds north from F), then Ball 3 (16 yds east from F). Or the shortest path is to go from F to Ball 1 to Ball 2 to Ball 3, or F to Ball 2 to Ball 3, or F to Ball 3 to Ball 2. Wait, the three balls: Ball 1 is 9 yds left of F, Ball 3 is 16 yds right of F, Ball 2 is 12 yds up from F. So the distance from F to Ball 1 is 9 yds, Ball 1 to Ball 2: using Pythagoras, \( \sqrt{9^{2}+12^{2}}=\sqrt{81 + 144}=\sqrt{225}=15 \) yds. Ball 2 to Ball 3: \( \sqrt{16^{2}+12^{2}}=\sqrt{256+144}=\sqrt{400}=20 \) yds. But the shortest path: start at F, go to Ball 1 (9 yds), then to Ball 2 (15 yds), then to Ball 3 (20 yds)? No, wait, the total distance if we go F - Ball1 - Ball2 - Ball3: 9+15 + 20=44. But wait, the distance from F to Ball3 is 16 yds, F to Ball2 is 12 yds, and Ball1 - Ball3 is 25 yds, Ball1 - Ball2 is 15 yds, Ball2 - Ball3 is 20 yds. Wait, the triangle formed by Ball1, F, Ball3 is a straight line (since Ball1 - F - Ball3 is horizontal), length 9 + 16 = 25 yds. Ball2 is 12 yds north of F, so the triangle Ball1 - Ball2 - Ball3 is a right triangle with legs 25 yds and 12 yds? No, Ball1 is at (- 9,0), F at (0,0), Ball3 at (16,0), Ball2 at (0,12). So the distance from Ball1 to Ball2 is 15, Ball2 to Ball3 is 20, Ball1 to Ball3 is 25. So the shortest path is F - Ball1 - Ball2 - Ball3: 9+15 + 20 = 44? Wait, no, or F - Ball2 - Ball3: 12+20 = 32? No, that's not right. Wait, the problem says "starting at point F, he picks up each ball. What is the shortest distance Felix can travel to pick up all three croquet balls, ending at the point where he picks up the third ball." So the possible paths:

1. F -> Ball1 -> Ball2 -> Ball3: distance = 9 (F to Ball1)+15 (Ball1 to Ball2)+20 (Ball2 to Ball3)=44

1. F -> Ball2 -> Ball1 -> Ball3: distance = 12 (F to Ball2)+15 (Ball2 to Ball1)+25 (Ball1 to Ball3)=52

1. F -> Ball2 -> Ball3 -> Ball1: distance = 12 (F to Ball2)+20 (Ball2 to Ball3)+25 (Ball3 to Ball1)=57

1. F -> Ball3 -> Ball2 -> Ball1: distance = 16 (F to Ball3)+20 (Ball3 to Ball2)+15 (Ball2 to Ball1)=51

1. F -> Ball3 -> Ball1 -> Ball2: distance = 16 (F to Ball3)+25 (Ball3 to Ball1)+15 (Ball1 to Ball2)=56

1. F -> Ball1 -> Ball3 -> Ball2: distance = 9 (F to Ball1)+25 (Ball1 to Ball3)+12 (Ball3 to Ball2)? No, Ball3 to Ball2 is 20, not 12. Wait, I think the correct approach is that the three balls form a right triangle with legs 12 yds (vertical) and \( 9 + 16=25 \) yds (horizontal). So the shortest path is the perimeter of the triangle? No, wait, starting at F, to pick up all three, the shortest path is to go from F to Ball1 (9 yds), Ball1 to Ball3 (25 yds), and Ball3 to Ball2 (20 yds)? No, that's not. Wait, no, the distance from F to Ball1 is 9, Ball1 to Ball2 is 15, Ball2 to Ball3 is 20. 9+15 + 20 = 44. But wait, the distance from F to Ball2 is 12, Ball2 to Ball3 is 20, and F to Ball3 is 16. 12+20+16? No, that's 48. Wait, I think I made a mistake. Let's use the triangle inequality. The three points: F(0,0), Ball1(-9,0), Ball2(0,12), Ball3(16,0). So the distance from F to Ball1: 9, F to Ball2:12, F to Ball3:16. Ball1 to Ball2:15, Ball1 to Ball3:25, Ball2 to Ball3:20. The shortest path to visit all three points starting at F and ending at the third: the minimal path is F - Ball1…

Answer:

C. 15

Problem 16