Pythagorean Theorem Worksheets - Free Printable

The image you've uploaded is a Pythagorean Theorem Word Problems Worksheet (Math Worksheet 20) with an answer key provided. It contains four real-world problems that require the use of the Pythagorean Theorem, which states:

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

Where:
- $a$ and $b$ are the legs of a right triangle,
- $c$ is the hypotenuse (the longest side, opposite the right angle).

Let’s go through each problem, explain the solution, and verify the answers.

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🔹 Problem 1: Bottles on a Shelf

Problem:
There are 2 bottles that are beside each other that are 28 cm and 19 cm tall. The bottles are 7 cm apart. How far apart are the centers of the bottles?

Solution:

We can model this as a right triangle:
- The vertical difference in height between the tops of the bottles: $28 - 19 = 9$ cm.
- The horizontal distance between the bottles: 7 cm.
- We want to find the distance between the centers of the bottles, which is the hypotenuse of a right triangle with legs 9 cm and 7 cm.

Using the Pythagorean Theorem:

$$
a^2 + b^2 = c^2 \\
7^2 + 9^2 = c^2 \\
49 + 81 = c^2 \\
130 = c^2 \\
c = \sqrt{130} \approx 11.4 \text{ cm}
$$

✔ Answer: 11.4 cm — Correct.

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🔹 Problem 2: Plant Hung from Ceiling

Problem:
Keisha bought a plant and hung it 580 mm high up from the ground to the base of the pot. The base parts of the pot and bookshelf are 700 mm diagonally apart. What is the horizontal distance between them?

Solution:

This is a right triangle where:
- Vertical leg ($a$): 580 mm (height),
- Hypotenuse ($c$): 700 mm (diagonal distance),
- We need to find the horizontal leg ($b$).

Use the Pythagorean Theorem:

$$
a^2 + b^2 = c^2 \\
580^2 + b^2 = 700^2 \\
336400 + b^2 = 490000 \\
b^2 = 490000 - 336400 = 153600 \\
b = \sqrt{153600} \approx 391.92 \text{ mm}
$$

✔ Answer: 391.92 mm — Correct.

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🔹 Problem 3: Fishing Boat Rope

Problem:
The pole of a fishing boat is supported by a sturdy rope that extends from the top of the pole to the deck. The pole is 20 feet tall and the rope attached to the deck is 15 feet away from the base of the pole. How long is the rope?

Solution:

Right triangle:
- Vertical leg ($a$): 20 ft,
- Horizontal leg ($b$): 15 ft,
- Hypotenuse ($c$): length of the rope.

$$
a^2 + b^2 = c^2 \\
20^2 + 15^2 = c^2 \\
400 + 225 = c^2 \\
625 = c^2 \\
c = \sqrt{625} = 25 \text{ feet}
$$

✔ Answer: 25 feet — Correct.

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🔹 Problem 4: Kite Flying

Problem:
Fenis went to a level field to fly a kite. He let out 204 meters of string and tied it to a tree. Then, he walked out on the field until she was directly under the kite, which was 182 meters from the tree. How high was the kite from the ground?

Solution:

Right triangle:
- Hypotenuse ($c$): 204 m (string length),
- Horizontal leg ($b$): 182 m (distance from tree to point under kite),
- Vertical leg ($a$): height of kite (what we're solving for).

$$
a^2 + b^2 = c^2 \\
a^2 + 182^2 = 204^2 \\
a^2 + 33124 = 41616 \\
a^2 = 41616 - 33124 = 8492 \\
a = \sqrt{8492} \approx 92.15 \text{ meters}
$$

✔ Answer: 92.15 meters — Correct.

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✔ Final Summary:

All four problems are solved correctly using the Pythagorean Theorem. Each scenario involves identifying the right triangle formed by the given measurements and applying $a^2 + b^2 = c^2$. The answers have been rounded to the nearest tenth (or appropriate decimal place), as instructed.

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📌 Key Takeaways:

- Always identify the right triangle and label the sides.
- Determine which side is the hypotenuse (longest side).
- Plug values into $a^2 + b^2 = c^2$.
- Solve algebraically and round appropriately.

Let me know if you'd like a printable version or further explanation!