Question

which lengths could represent the side lengths of a right triangle?
i. 7, 8, \sqrt{167}
ii. 5, 7, \sqrt{74}
iii. 3, 13, \sqrt{173}
choose the correct answer below.
a. i, ii, and iii
b. ii only
c. i and ii only
click to select your answer.

Explanation:

Step1: Test Set I with Pythagoras

Check if $7^2 + 8^2 = (\sqrt{167})^2$

$$\begin{aligned} 7^2 + 8^2 &= 49 + 64 = 113 \\ (\sqrt{167})^2 &= 167 \end{aligned}$$

$113
eq 167$, so Set I fails.

Step2: Test Set II with Pythagoras

Check if $5^2 + 7^2 = (\sqrt{74})^2$

$$\begin{aligned} 5^2 + 7^2 &= 25 + 49 = 74 \\ (\sqrt{74})^2 &= 74 \end{aligned}$$

$74 = 74$, so Set II passes.

Step3: Test Set III with Pythagoras

Check if $3^2 + 13^2 = (\sqrt{173})^2$

$$\begin{aligned} 3^2 + 13^2 &= 9 + 169 = 178 \\ (\sqrt{173})^2 &= 173 \end{aligned}$$

$178
eq 173$, so Set III fails.

Answer:

B. II only