Question

question 2
suppose triangles p, q, and r have sides with the given measurements.

• triangle p: 12, 24, and 30
• triangle q: 9, 40, and 41
• triangle r: 5, 18, and 21

which triangle is a right triangle? explain your reasoning.

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, $a^2 + b^2 = c^2$, where $c$ is the longest side.

Step2: Test Triangle P

Check $12^2 + 24^2$ vs $30^2$

$$\begin{align} 12^2 + 24^2 &= 144 + 576 = 720 \\ 30^2 &= 900 \end{align}$$

$720
eq 900$, so not a right triangle.

Step3: Test Triangle Q

Check $9^2 + 40^2$ vs $41^2$

$$\begin{align} 9^2 + 40^2 &= 81 + 1600 = 1681 \\ 41^2 &= 1681 \end{align}$$

$1681 = 1681$, so this is a right triangle.

Step4: Test Triangle R (verification)

Check $5^2 + 18^2$ vs $21^2$

$$\begin{align} 5^2 + 18^2 &= 25 + 324 = 349 \\ 21^2 &= 441 \end{align}$$

$349
eq 441$, so not a right triangle.

Answer:

Triangle Q (with sides 9, 40, and 41) is a right triangle, as it satisfies the Pythagorean theorem: $9^2 + 40^2 = 41^2$.