Question

applying the pythagorean theorem
use the pythagorean theorem to help you answer the following questions. show all work and round to the nearest tenth when necessary.

1. layne rode his bike from point a to b by using cherry street. how much further would his trip have been if he took orange drive and peach avenue instead?

cherry st. 85 yd
peach ave. 77 yd
orange dr.

1. ashton left his house and ran 4 miles east and then 3 miles north. he then took the diagonal path back home. if he burned 105 calories every mile that he ran, how many total calories did he burn on his run?
2. the blackburn family has a square field where they keep their cattle. the area of the field is 40,000 $ft^{2}$, and mr. blackburn wants to put a fence diagonally through the field. what should the length of the fence be?
3. martin needs to place caution tape across both diagonals of a broken rectangular door in his store. if the dimensions of the door are 3 feet by 7 feet, how many feet of caution tape will he need?
4. tony is building a doghouse, and the front view of the roof is an isosceles triangle as shown below. what is the height of the roof?

29 inches
40 inches
$x$

1. if the area of square 2 is 225 $units^{2}$, and the perimeter of square 1 is 100 units, what is the area of square 3?

Explanation:

Problem 1

Step1: Find length of Orange Dr.

Use Pythagorean theorem: $a^2 + b^2 = c^2$. Let $a=77$, $c=85$, solve for $a$ (Orange Dr.):
$$a = \sqrt{85^2 - 77^2} = \sqrt{7225 - 5929} = \sqrt{1296} = 36 \text{ yd}$$

Step2: Calculate total alternate trip length

Sum Orange Dr. and Peach Ave.:
$$36 + 77 = 113 \text{ yd}$$

Step3: Find difference in trip lengths

Subtract Cherry St. length from alternate trip length:
$$113 - 85 = 28 \text{ yd}$$

Problem 2

Step1: Find diagonal distance

Use Pythagorean theorem for 4mi east, 3mi north:
$$d = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ mi}$$

Step2: Calculate total miles run

Sum all segments:
$$4 + 3 + 5 = 12 \text{ mi}$$

Step3: Compute total calories burned

Multiply total miles by calories per mile:
$$12 \times 105 = 1260 \text{ calories}$$

Problem 3

Step1: Find side length of square

Area $A = s^2$, solve for $s$:
$$s = \sqrt{40000} = 200 \text{ ft}$$

Step2: Calculate diagonal length

Use Pythagorean theorem for square:
$$d = \sqrt{200^2 + 200^2} = \sqrt{80000} \approx 282.8 \text{ ft}$$

Problem 4

Step1: Find one diagonal length

Use Pythagorean theorem for 3ft×7ft rectangle:
$$d = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \approx 7.62 \text{ ft}$$

Step2: Calculate total tape length

Multiply single diagonal by 2:
$$2 \times 7.62 = 15.2 \text{ ft}$$

Problem 5

Step1: Split base into two equal parts

Divide 40in base:
$$\frac{40}{2} = 20 \text{ in}$$

Step2: Calculate roof height

Use Pythagorean theorem for right triangle:
$$x = \sqrt{29^2 - 20^2} = \sqrt{841 - 400} = \sqrt{441} = 21 \text{ in}$$

Problem 6

Step1: Find side length of square 1

Perimeter $P=4s$, solve for $s$:
$$s = \frac{100}{4} = 25 \text{ units}$$

Step2: Find area of square 1

$$A_1 = 25^2 = 625 \text{ units}^2$$

Step3: Find area of square 3

Use Pythagorean theorem for squares: $A_1 = A_2 + A_3$, solve for $A_3$:
$$A_3 = 625 - 225 = 400 \text{ units}^2$$

Answer:

1. 28 yd
2. 1260 calories
3. 282.8 ft
4. 15.2 ft
5. 21 inches
6. 400 units²