We are given several math problems to solve, including simplifying radicals, finding the length of the third side of right triangles, and identifying sets of numbers that could represent the sides of a right triangle.

1. Problem Description

We are given several math problems to solve, including simplifying radicals, finding the length of the third side of right triangles, and identifying sets 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

1

6. Let the third side be $c$.

Using the Pythagorean theorem:

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

$12^2 + 16^2 = c^2$

$144 + 256 = c^2$

$400 = c^2$

$c = \sqrt{400} = 20$

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

2

5. Let the other leg be $b$.

Using the Pythagorean theorem:

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

$15^2 + b^2 = 25^2$

$225 + b^2 = 625$

$b^2 = 625 - 225$

$b^2 = 400$

$b = \sqrt{400} = 20$

Problem 5: Find the hypotenuse of a right triangle with legs of length 5 cm and 2 cm. Let the hypotenuse be $c$.

Using the Pythagorean theorem:

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

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

$25 + 4 = c^2$

$29 = c^2$

$c = \sqrt{29} \approx 5.4$ (rounded to the nearest tenth)

Problem 6: A 39-foot ladder is placed against a building. The bottom of the ladder is 33 feet from the bottom of the building. How tall is the building? Let the height of the building be $b$.

Using the Pythagorean theorem:

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

$33^2 + b^2 = 39^2$

$1089 + b^2 = 1521$

$b^2 = 1521 - 1089$

$b^2 = 432$

$b = \sqrt{432} \approx 20.8$ (rounded to the nearest tenth)

Problem 7: Which of the following sets of numbers could represent the sides of a right triangle? We need to check which set satisfies the Pythagorean theorem $a^2 + b^2 = c^2$, where $c$ is the largest number in the set.

A. {20, 21, 28}: $20^2 + 21^2 = 400 + 441 = 841$. $28^2 = 784$. $841 \neq 784$.

B. {24, 32, 39}: $24^2 + 32^2 = 576 + 1024 = 1600$. $39^2 = 1521$. $1600 \neq 1521$.

C. {31, 60, 68}: $31^2 + 60^2 = 961 + 3600 = 4561$. $68^2 = 4624$. $4561 \neq 4624$.

D. {6, 8, 10}: $6^2 + 8^2 = 36 + 64 = 100$. $10^2 = 100$. $100 = 100$. This set represents 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}