Pythagorean Theorem practice problems on a math worksheet.

It looks like you've uploaded an image containing a worksheet with three geometry problems involving Pythagorean Theorem applications. I’ll solve each problem step by step and explain the reasoning.

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Problem 1:

> What is the approximate height, *h*, of the kite and the string from the top of the tree to where it meets on the ground? Use *d* = 40 ft to determine the height.

Diagram Description:
A right triangle is shown:
- One leg (horizontal) = 40 ft (distance from tree to where string touches ground).
- Hypotenuse = 50 ft (length of string).
- Vertical leg = *h* (height of kite).

We are to find *h*, using the Pythagorean Theorem:

$$
a^2 + b^2 = c^2
$$

Let:
- $ a = h $ (unknown)
- $ b = 40 $ ft
- $ c = 50 $ ft

$$
h^2 + 40^2 = 50^2 \\
h^2 + 1600 = 2500 \\
h^2 = 2500 - 1600 = 900 \\
h = \sqrt{900} = 30
$$

✔ Answer: A. 30 feet

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Problem 2:

> Sean is building a triangular sandbox for his little brother. He has already built two sides: one side is 8 feet long, and the other side is 4 feet long. How long should the last side be?

Important Note: This is ambiguous unless we know it's a right triangle. But the diagram shows a right triangle, so we assume it’s a right triangle with legs 8 ft and 4 ft.

So we use the Pythagorean Theorem to find the hypotenuse:

$$
c^2 = a^2 + b^2 \\
c^2 = 8^2 + 4^2 = 64 + 16 = 80 \\
c = \sqrt{80} \approx 8.94 \text{ ft}
$$

But the options are:
A. 1 foot
B. 7 feet
C. 8 feet
D. 9 feet

The closest is D. 9 feet, but let's double-check if this is really about the hypotenuse.

Wait — could it be that the last side is not the hypotenuse? Could it be a leg?

But the diagram shows a right triangle with two legs labeled, and the third side (hypotenuse) missing. So yes, it's asking for the hypotenuse.

Since $ \sqrt{80} \approx 8.94 $, which rounds to 9 feet, the best choice is:

✔ Answer: D. 9 feet

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Problem 3:

> On a camping trip, Trina is going to build a shelter by leaning a tarp up against a wall. She wants her sleeping bag to be perpendicular to the wall. Her sleeping bag is 10 feet long and the wall is 8 feet high. How long must the tarp be?

Diagram Description:
A right triangle:
- Vertical leg = 8 ft (wall height)
- Horizontal leg = 10 ft (sleeping bag length, perpendicular to wall)
- Hypotenuse = ? (length of tarp)

This is again a right triangle. We need to find the hypotenuse:

$$
c^2 = 8^2 + 10^2 = 64 + 100 = 164 \\
c = \sqrt{164} \approx 12.8 \text{ ft}
$$

Now look at the choices:
A. 10 feet
B. 12 feet
C. 13 feet
D. 160 feet

$ \sqrt{164} \approx 12.8 $, so the closest whole number is 13 feet.

✔ Answer: C. 13 feet

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✔ Final Answers:

1. A. 30 feet
2. D. 9 feet
3. C. 13 feet

Let me know if you'd like a visual explanation or want these solved differently!