We are asked to solve problems 5, 6, and 7 from the "Right Triangles Test" image. Problem 5: One leg of a right triangle measures 5 cm and the other leg measures 2 cm. We need to find the measure of the hypotenuse, rounded to the nearest tenth. Problem 6: A 39-foot ladder is placed against the top of a building, and the bottom of the ladder is 33 feet from the bottom of the building. We need to find the height of 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? The options are A. {20, 21, 28}, B. {24, 32, 39}, C. {31, 60, 68}, and D. {6, 8, 10}.

1. Problem Description

We are asked to solve problems 5, 6, and 7 from the "Right Triangles Test" image.

Problem 5: One leg of a right triangle measures 5 cm and the other leg measures 2 cm. We need to find the measure of the hypotenuse, rounded to the nearest tenth.

Problem 6: A 39-foot ladder is placed against the top of a building, and the bottom of the ladder is 33 feet from the bottom of the building. We need to find the height of 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? The options are A. {20, 21, 28}, B. {24, 32, 39}, C. {31, 60, 68}, and D. {6, 8, 10}.

2. Solution Steps

Problem 5:

Let $a$ and $b$ be the lengths of the legs and $c$ be the length of the hypotenuse of a right triangle.

Pythagorean theorem:

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

Given $a=5$ and $b=2$, we have $5^2 + 2^2 = c^2$, which means $25 + 4 = c^2$.

So $c^2 = 29$, and $c = \sqrt{29}$.

$\sqrt{29} \approx 5.385$.

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

Problem 6:

Let $h$ be the height of the building, and $d$ be the distance from the bottom of the building to the bottom of the ladder. The length of the ladder is $l$.

Pythagorean theorem:

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

Given $l=39$ and $d=33$, we have $h^2 + 33^2 = 39^2$, which means $h^2 + 1089 = 1521$.

So $h^2 = 1521 - 1089 = 432$, and $h = \sqrt{432}$.

$\sqrt{432} \approx 20.7846$.

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

Problem 7:

A set of numbers {a, b, c} can represent the sides of a right triangle if the Pythagorean theorem holds, i.e., $a^2 + b^2 = c^2$ for some ordering of the numbers. We need to check each option.

A. {20, 21, 28}: $20^2 + 21^2 = 400 + 441 = 841$. $28^2 = 784$. Since $841 \neq 784$, this is not a right triangle.

B. {24, 32, 39}: $24^2 + 32^2 = 576 + 1024 = 1600$. $39^2 = 1521$. Since $1600 \neq 1521$, this is not a right triangle.

C. {31, 60, 68}: $31^2 + 60^2 = 961 + 3600 = 4561$. $68^2 = 4624$. Since $4561 \neq 4624$, this is not a right triangle.

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

3. Final Answer

Problem 5: The hypotenuse is approximately 5.4 cm.

Problem 6: The height of the building is approximately 20.8 feet.

Problem 7: The set of numbers that could represent the three sides of a right triangle is D. {6, 8, 10}.