Question

find the length of side ( x ) to the nearest tenth.
answer attempt 1 out of 2
( x = )
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Explanation:

Step1: Identify triangle type

This is a 30-60-90 right triangle. The hypotenuse is 3.

Step2: Recall 30-60-90 ratios

In a 30-60-90 triangle, the side opposite 30° is half the hypotenuse, and the side opposite 60° is $\frac{\sqrt{3}}{2}$ times the hypotenuse. Here, \( x \) is opposite 30°? Wait, no: wait, angles are 30°, 60°, 90°. The side labeled 3 is opposite the 90°? Wait, no, the right angle is at the bottom, so the side of length 3 is the hypotenuse (opposite the right angle). Then, the side \( x \) is opposite the 30° angle? Wait, no: the angles are 30°, 60°, 90°. So the side opposite 30° is half the hypotenuse. Wait, hypotenuse is 3. So side opposite 30° is \( \frac{3}{2} = 1.5 \)? Wait, no, wait: let's label the triangle. Let’s call the right angle \( C \), 30° angle \( A \), 60° angle \( B \). Then side opposite \( A \) (30°) is \( BC = x \), side opposite \( B \) (60°) is \( AC \), and hypotenuse \( AB = 3 \). In 30-60-90 triangle, \( \sin(30^\circ) = \frac{BC}{AB} \), so \( BC = AB \times \sin(30^\circ) \). \( \sin(30^\circ) = 0.5 \), so \( x = 3 \times 0.5 = 1.5 \)? Wait, but wait, maybe I mixed up. Wait, alternatively, \( \cos(60^\circ) = \frac{BC}{AB} \), since \( \cos(60^\circ) = 0.5 \), so \( x = 3 \times 0.5 = 1.5 \). Or using \( \sin(60^\circ) \) for the other side, but here we need \( x \), which is opposite 30°, so \( \sin(30^\circ) = \frac{x}{3} \), so \( x = 3 \times \sin(30^\circ) = 3 \times 0.5 = 1.5 \). Wait, but that seems too simple. Wait, maybe I made a mistake. Wait, no, 30-60-90 triangle: sides are in ratio \( 1 : \sqrt{3} : 2 \), where the side opposite 30° is 1, opposite 60° is \( \sqrt{3} \), hypotenuse 2. So if hypotenuse is 3, then the side opposite 30° is \( \frac{3}{2} = 1.5 \), which is 1.5. So to the nearest tenth, 1.5.

Step3: Verify

Alternatively, using cosine: \( \cos(60^\circ) = \frac{x}{3} \), since \( \cos(60^\circ) = 0.5 \), so \( x = 3 \times 0.5 = 1.5 \). Yep, that's correct.

Answer:

\( x = 1.5 \)