We need to solve four geometry problems related to right triangles: simplifying radicals, finding the length of a side given other sides (using the Pythagorean theorem), and identifying a set of numbers that could represent the sides of a right triangle.

1. Problem Description

We need to solve four geometry problems related to right triangles: simplifying radicals, finding the length of a side given other sides (using the Pythagorean theorem), and identifying a set of numbers that could represent the sides of a right triangle.

2. Solution Steps

Problem 1: Simplify $\sqrt{125}$

$\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}$

Problem 2: Simplify $\sqrt{100}$

$\sqrt{100} = 10$

Problem 3: Find the length of the third side of a right triangle with legs of length 12 and

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6. We use the Pythagorean theorem: $a^2 + b^2 = c^2$.

In this case, $12^2 + 16^2 = c^2$.

$144 + 256 = c^2$

$400 = c^2$

$c = \sqrt{400} = 20$

Problem 4: Find the length of the third side of a right triangle with one leg of length 15 and hypotenuse of length

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5. Let $a$ be the unknown leg, $b=15$, and $c=25$ be the hypotenuse.

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

$a^2 + 15^2 = 25^2$

$a^2 + 225 = 625$

$a^2 = 625 - 225$

$a^2 = 400$

$a = \sqrt{400} = 20$

Problem 5: Find the length of the hypotenuse of a right triangle with legs of lengths 5 cm and 2 cm.

$a = 5$, $b = 2$, and $c$ is the hypotenuse.

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

$5^2 + 2^2 = c^2$

$25 + 4 = c^2$

$29 = c^2$

$c = \sqrt{29} \approx 5.385$

Round to the nearest tenth: $c \approx 5.4$

Problem 6: A 39-foot ladder leans against a building. The bottom of the ladder is 33 feet from the bottom of the building. How tall is the building?

Let $a$ be the height of the building, $b = 33$ feet, and $c = 39$ feet (the ladder).

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

$a^2 + 33^2 = 39^2$

$a^2 + 1089 = 1521$

$a^2 = 1521 - 1089$

$a^2 = 432$

$a = \sqrt{432} \approx 20.7846$

Round to the nearest tenth: $a \approx 20.8$

Problem 7: Which of the following sets of numbers could represent the three sides of a right triangle?

A. {20, 21, 28}: $20^2 + 21^2 = 400 + 441 = 841$. $28^2 = 784$. Not a right triangle.

B. {24, 32, 39}: $24^2 + 32^2 = 576 + 1024 = 1600$. $39^2 = 1521$. Not a right triangle.

C. {31, 60, 68}: $31^2 + 60^2 = 961 + 3600 = 4561$. $68^2 = 4624$. Not a right triangle.

D. {6, 8, 10}: $6^2 + 8^2 = 36 + 64 = 100$. $10^2 = 100$. This is a right triangle.

3. Final Answer

Problem 1: $5\sqrt{5}$

Problem 2: $10$

Problem 3: $20$

Problem 4: $20$

Problem 5: $5.4$ cm

Problem 6: $20.8$ feet

Problem 7: D. {6, 8, 10}