Question

determine whether a triangle with the given side lengths is a right triangle.

side lengths	right triangle	not a right triangle	not enough information
(b) 7, 12, 14	○	○	○
(c) 4, 8, 9	○	○	○
(d) 11, 13, 17	○	○	○

Explanation:

Step1: Apply Pythagorean theorem to (a)

Check if $12^2 + 35^2 = 37^2$

$$\begin{align*} 12^2 + 35^2 &= 144 + 1225 = 1369\\ 37^2 &= 1369 \end{align*}$$

Step2: Apply Pythagorean theorem to (b)

Check if $7^2 + 12^2 = 14^2$

$$\begin{align*} 7^2 + 12^2 &= 49 + 144 = 193\\ 14^2 &= 196 \end{align*}$$

Step3: Apply Pythagorean theorem to (c)

Check if $4^2 + 8^2 = 9^2$

$$\begin{align*} 4^2 + 8^2 &= 16 + 64 = 80\\ 9^2 &= 81 \end{align*}$$

Step4: Apply Pythagorean theorem to (d)

Check if $11^2 + 13^2 = 17^2$

$$\begin{align*} 11^2 + 13^2 &= 121 + 169 = 290\\ 17^2 &= 289 \end{align*}$$

Answer:

(a) Right triangle
(b) Not a right triangle
(c) Not a right triangle
(d) Not a right triangle