Question

1. andre and clare are discussing the triangle ( abc ) that has a right angle at ( c ) and a hypotenuse of length 15 units. andre thinks the triangle could have legs that are 9 and 12 units long. clare thinks angle ( b ) could be ( 20^circ ) and then side ( bc ) would be 14.1 units long. do you agree with either of them? explain or show your reasoning.

Explanation:

Step1: Verify Andre's claim

Check Pythagorean theorem:
$$9^2 + 12^2 = 81 + 144 = 225$$
$$15^2 = 225$$
Since $9^2 + 12^2 = 15^2$, the legs satisfy the theorem.

Step2: Verify Clare's claim

Use cosine for $\angle B$:
$\cos(B) = \frac{BC}{AB}$, so $BC = AB \cos(B)$
$$BC = 15 \cos(20^\circ) \approx 15 \times 0.9397 \approx 14.0955$$
This rounds to 14.1 units.

Answer:

I agree with both Andre and Clare. Andre's legs satisfy the Pythagorean theorem for the given hypotenuse, and Clare's calculation for side $BC$ using trigonometry is accurate when $\angle B = 20^\circ$.