Question

Is (5, 7, 74−−√
) a Pythagorean triple? (1 point)
Responses

no
no

yes

Given that (20, x, 29) is a Pythagorean triple and x < 29, what is the value of x?
(1 point)
Responses

21
21

18
18

19
19

22

A 24-foot-tall tree casts a shadow that is 10 feet long. What is the distance from the top of the tree to the top of its shadow? Apply Pythagorean triple to solve the problem. (1 point)
Responses

26
26

27
27

100
100

25

Tyrese buys potting soil every Janurary to plant vegetables in his garden. This year he bought 15 25-pound bags of potting soil. He must push them up a 6-foot-high ramp to his truck. The horizontal distance from the base of the ramp to the truck is 8 feet. Apply Pythagorean triple to find the length of the ramp. (1 point)
Responses

11
11

10
10

9
9

15

Given that (11, 60, 61) is a Pythagorean triple, use (11, 60, 61) to find a new Pythagorean triple that contains 183. (1 point)
The new Pythagorean triple is ( blank
, blank
, 183)


A cat climbs on top of a house and looks down at you. You are standing 12 feet away from the house and there is an 18-foot diagonal distance between you and the cat. Using the inverse of sine function, find the angle between the side of the house and the cat's line of sight looking down at you. Round your answer to the nearest whole degree. (1 point)
Responses

41
41

40
40

42
42

34

Oscar uses a board to form a walking plank across a small ditch that has uneven sides. The board is 6 feet in length and stretches across the entire gap. The higher side of the ditch is 3 feet above the other side. Using the inverse of sine, what is the approximate angle of elevation formed between the board and the lower side of the ditch? (1 point)
Responses

31
31

29
29

15
15

30


Jaclynn is building a cat house for her cats. She deisgns a ramp from one level to the other for them to climb. The ramp lenght is 4 feet, and the difference between the two levels is 2.5 feet. Using the inverse sine function, what is the appoximate angle of elevation formed by the rampl? Round your answer to the nearest whole degree.(1 point)
Responses

36
36

38
38

37
37

39


Meekah is looking at a bird perched on top of a street light. Her line of sight, which is the diagonal distance to the top of the street light, is 16 feet, and she is standing 8 feet from the base of the street light. Use the inverse of cosine to find the angle of elevation with which she is looking at the bird. Round your answer to the nearest whole degree.

(1 point)
Responses

55
55

60
60

50
50

40
40

Answer 1

Let's tackle each question one by one:

1. **Is (5, 7, √74) a Pythagorean triple?**
- For a Pythagorean triple, the equation \(a^2 + b^2 = c^2\) must hold.
- We can check:
\[
5^2 + 7^2 = 25 + 49 = 74
\]
\[
√74^2 = 74
\]
- Since \(74 = 74\), (5, 7, √74) **is** a Pythagorean triple.

**Response:** yes

2. **Given that (20, x, 29) is a Pythagorean triple and x < 29, what is the value of x?**
- Using the Pythagorean theorem:
\[
20^2 + x^2 = 29^2
\]
\[
400 + x^2 = 841
\]
\[
x^2 = 841 - 400 = 441
\]
\[
x = \sqrt{441} = 21
\]

**Response:** 21

3. **A 24-foot tall tree casts a shadow that is 10 feet long. What is the distance from the top of the tree to the top of its shadow?**
- Use the Pythagorean theorem:
\[
24^2 + 10^2 = d^2
\]
\[
576 + 100 = d^2
\]
\[
d^2 = 676
\]
\[
d = \sqrt{676} = 26
\]

**Response:** 26

4. **Tyrese's ramp problem:**
- Given the height of 6 feet and a horizontal distance of 8 feet, find the length of the ramp using Pythagorean theorem:
\[
6^2 + 8^2 = r^2
\]
\[
36 + 64 = r^2
\]
\[
r^2 = 100
\]
\[
r = \sqrt{100} = 10
\]

**Response:** 10

5. **Find a new Pythagorean triple that contains 183 using (11, 60, 61):**
- The relationship can be found using a scaling factor. The scaling factor to get from 61 to 183 is:
\[
k = \frac{183}{61} = 3
\]
- Thus, the new triple will be:
\[
(11 \times 3, 60 \times 3, 183) = (33, 180, 183)
\]

**Response:** The new Pythagorean triple is (33, 180, 183)

6. **Using inverse sine for angle when standing 12 feet from the house and with an 18-foot diagonal distance:**
- The height of the house (opposite side) is 18 feet,
\[
\sin(\theta) = \frac{opposite}{hypotenuse} = \frac{18}{18}
\]
Here, the formula to find the angle is:
\[
\sin(\theta) = \frac{12}{18} \implies \theta = \arcsin\left(\frac{12}{18}\right)
\]
\(\theta \approx 41^\circ \)

**Response:** 41

7. **Using inverse sine for the ramp and 3-foot height:**
- Considering the opposite side is 3 feet and the hypotenuse is 6 feet,
\[
\sin(\theta) = \frac{3}{6} \implies \theta = \arcsin\left(\frac{1}{2}\right)
\]
\(\theta \approx 30^\circ \)

**Response:** 30

8. **Using inverse sine for ramp of 4 feet with a height difference of 2.5 feet:**
\[
\sin(\theta) = \frac{2.5}{4}
\]
Find the angle:
\[
\theta = \arcsin\left(\frac{2.5}{4}\right) \approx 38^°
\]

**Response:** 38

9. **Using inverse cosine with 16 feet diagonal and standing 8 feet away:**
\[
\cos(\theta) = \frac{8}{16} = \frac{1}{2}
\]
\(\theta = \cos^{-1}\left(\frac{1}{2}\right) \approx 60^\circ \)

**Response:** 60

Feel free to ask any follow-up questions!