Worksheet on the Pythagorean Theorem with real-world application problems.

It looks like you've uploaded a worksheet titled "Pythagorean Theorem" with several problems related to the Pythagorean Theorem, which states:

> In a right triangle:
> $$
> a^2 + b^2 = c^2
> $$
> where $a$ and $b$ are the legs, and $c$ is the hypotenuse.

Let’s go through each problem step by step and solve them.

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Problem 1:

> Mark is in his room looking at a wall. He draws 30 inches down the wall and then 12 inches over. How far did he travel? Round your answer to the nearest tenth.

This describes a right triangle:
- One leg: 30 inches (down)
- Other leg: 12 inches (over)
- We want the hypotenuse (distance traveled)

Using the Pythagorean Theorem:
$$
c^2 = a^2 + b^2 = 30^2 + 12^2 = 900 + 144 = 1044
$$
$$
c = \sqrt{1044} \approx 32.3 \text{ inches}
$$

✔ Answer: Approximately 32.3 inches

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Problem 2:

> Mr. Whiskers has an outdoor kitchen area shaped like a right triangle. It is 6 feet high and 8 feet wide. What is the radius of the circle that would fit exactly in the triangle?

Wait — this is tricky. The question says "the radius of the circle that would fit exactly in the triangle." That likely refers to the incircle (circle inscribed in the triangle).

For a right triangle, the radius $r$ of the incircle is given by:
$$
r = \frac{a + b - c}{2}
$$
where $a$ and $b$ are the legs, and $c$ is the hypotenuse.

Given:
- $a = 6$ ft
- $b = 8$ ft
- Find $c$:
$$
c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ ft}
$$

Now compute:
$$
r = \frac{6 + 8 - 10}{2} = \frac{4}{2} = 2 \text{ ft}
$$

✔ Answer: The radius is 2 feet

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Problem 3:

> You want to construct a ramp with a height of 3 feet and a length of 5 feet. If you use this as the hypotenuse, what should be the base of the ramp?

We are told:
- Height (one leg): $a = 3$ ft
- Hypotenuse: $c = 5$ ft
- Find the base ($b$)

Use:
$$
a^2 + b^2 = c^2 \\
3^2 + b^2 = 5^2 \\
9 + b^2 = 25 \\
b^2 = 16 \\
b = \sqrt{16} = 4 \text{ ft}
$$

✔ Answer: The base should be 4 feet

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Problem 4:

> A 15-foot ladder leans against a building. The top of the ladder is 12 feet up the side of the building. How far from the base of the building is the bottom of the ladder?

This is a classic right triangle setup:
- Hypotenuse (ladder): $c = 15$ ft
- Vertical leg (height): $a = 12$ ft
- Find horizontal leg ($b$) — distance from wall

$$
a^2 + b^2 = c^2 \\
12^2 + b^2 = 15^2 \\
144 + b^2 = 225 \\
b^2 = 81 \\
b = \sqrt{81} = 9 \text{ ft}
$$

✔ Answer: The bottom of the ladder is 9 feet from the building

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Problem 5:

> Jack is building a patio for his apartment. The shape is a 3 meter by 4 meter rectangle. What is the diagonal length of the patio?

A rectangle's diagonal forms a right triangle with sides 3 m and 4 m.

So:
$$
d^2 = 3^2 + 4^2 = 9 + 16 = 25 \\
d = \sqrt{25} = 5 \text{ meters}
$$

✔ Answer: The diagonal is 5 meters

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✔ Final Answers Summary:

1. 32.3 inches
2. 2 feet
3. 4 feet
4. 9 feet
5. 5 meters

Let me know if you'd like these explained visually or with diagrams!