Question

what is the value of x in the triangle? triangle with right angle, 30°, 60°, and one leg 15 options: a. 15√3, b. 7.5, c. 5, d. 5√3, e. 15

Explanation:

Step1: Identify triangle type

It's a 30-60-90 right triangle.

Step2: Relate side to angle

The side of length 15 is opposite the 60° angle; x is opposite the 30° angle.

Step3: Use 30-60-90 ratio

In such triangles, the side opposite 30° is half the hypotenuse, but alternatively, the side opposite 60° is $\sqrt{3}$ times the side opposite 30°. So $15 = x\sqrt{3}$ → $x = \frac{15}{\sqrt{3}} = 5\sqrt{3}$? Wait no, wait: Wait the side labeled 15 is adjacent to 30°? Wait no, let's check again: The angle at the top left is 30°, the right angle is top right, so the side of length 15 is adjacent to 30° and opposite 60°? Wait no, in the triangle, the angle of 30°: its adjacent side is x? No, wait the right angle is between x and the side of length 15? Wait no, the triangle has angles 30°, 60°, 90°, so sides are in ratio 1 : $\sqrt{3}$ : 2 (opposite 30°, 60°, 90°). Let's denote: side opposite 30° is S, opposite 60° is $S\sqrt{3}$, hypotenuse is 2S. The side labeled 15: which angle is it opposite? The 60° angle (since the 60° angle is at the bottom right, so its opposite side is the top side of length 15). So 15 = $S\sqrt{3}$? No, wait no: the 60° angle's opposite side is the side opposite to it, which is the side of length 15? Wait the bottom right angle is 60°, so its opposite side is the left side (hypotenuse? No). Wait maybe better to use trigonometry: tan(30°) = opposite/adjacent = x/15? No, tan(30°) = opposite over adjacent for 30° angle: opposite side is x, adjacent is 15? Wait tan(30°) = x/15 → x = 15 tan(30°) =15(1/√3)=5√3? But wait option D is 5√3. Wait wait I made a mistake earlier: let's confirm: angle at top left is 30°, so its opposite side is the side opposite to it, which is the side opposite 30°, which is the shorter leg. The side of length 15: is it adjacent to 30°? Yes, because the 30° angle has two sides: one is the hypotenuse, the other is the adjacent side of length 15. So cos(30°)=adjacent/hypotenuse=15/hypotenuse, but we need x. Alternatively, sin(30°)=opposite/hypotenuse=x/hypotenuse, and cos(30°)=15/hypotenuse. So sin(30°)/cos(30°)=tan(30°)=x/15 → x=15 tan(30°)=15(1/√3)=5√3. Oh right! So the correct answer is D. 5√3.

Step1: Confirm triangle type

It is a 30-60-90 right triangle.

Step2: Assign trigonometric ratio

For the 30° angle, $\tan(30^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{15}$.

Step3: Calculate x

$\tan(30^\circ) = \frac{1}{\sqrt{3}}$, so $x = 15 \times \frac{1}{\sqrt{3}} = 5\sqrt{3}$.

Answer:

B. 7.5