The image contains several right triangle problems. We need to solve problems 5, 6, and 7. Problem 5: One leg of a right triangle measures 5 cm and the other leg measures 2 cm. Find the length of the hypotenuse, 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, rounded to the nearest tenth of a foot? Problem 7: Which of the following sets of numbers could represent the three sides of a right triangle? A. {20, 21, 28} B. {24, 32, 39} C. {31, 60, 68} D. {6, 8, 10}

1. Problem Description

The image contains several right triangle problems. We need to solve problems 5, 6, and

7.

Problem 5: One leg of a right triangle measures 5 cm and the other leg measures 2 cm. Find the length of the hypotenuse, 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, rounded to the nearest tenth of a foot?

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

A. {20, 21, 28}

B. {24, 32, 39}

C. {31, 60, 68}

D. {6, 8, 10}

2. Solution Steps

Problem 5:

Let $a$ and $b$ be the lengths of the legs of the right triangle, and let $c$ be the length of the hypotenuse. We have $a = 5$ cm and $b = 2$ cm. By the Pythagorean theorem, we have $a^2 + b^2 = c^2$.

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

$c^2 = 5^2 + 2^2 = 25 + 4 = 29$

$c = \sqrt{29}$

$c \approx 5.385$

Rounding to the nearest tenth, $c \approx 5.4$ cm.

Problem 6:

Let $h$ be the height of the building, $l$ be the length of the ladder, and $d$ be the distance of the bottom of the ladder from the building. We are given that $l = 39$ feet and $d = 33$ feet. The ladder, building, and ground form a right triangle.

$h^2 + d^2 = l^2$

$h^2 = l^2 - d^2 = 39^2 - 33^2 = 1521 - 1089 = 432$

$h = \sqrt{432}$

$h = \sqrt{144 \times 3} = 12\sqrt{3} \approx 20.785$

Rounding to the nearest tenth, $h \approx 20.8$ feet.

Problem 7:

We need to check which set of numbers 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}: $6^2 + 8^2 = 36 + 64 = 100$. $10^2 = 100$. $100 = 100$.

3. Final Answer

Problem 5: 5.4 cm

Problem 6: 20.8 feet

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