Question

what is the value of x?
30°
13
x

Explanation:

Step1: Identify triangle type

It's a right - triangle with a 30° angle, so it's a 30 - 60 - 90 triangle. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite 30° is the shortest side (let's call it \(a\)), the side opposite 60° is \(a\sqrt{3}\), and the hypotenuse is \(2a\). Also, we can use trigonometric ratios. Let's use sine: \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), the hypotenuse is 13, and the side opposite 30° is \(x\)? Wait, no. Wait, the right angle is at the corner with \(x\) and the other side. Wait, let's re - label: the angle of 30° is at the top, the right angle is at the bottom - right, so the side of length 13 is the hypotenuse? Wait, no, in a right - triangle, the hypotenuse is opposite the right angle. Wait, the right angle is at the vertex with \(x\) and the other leg. So the side labeled 13: let's see, the angle of 30°: the side opposite 30° is \(x\), and the hypotenuse is the side opposite the right angle. Wait, no, let's use trigonometry. Let's denote the right - angle vertex as \(C\), the 30° vertex as \(A\), and the other vertex as \(B\). So \(\angle A = 30^{\circ}\), \(\angle C=90^{\circ}\), side \(AB = 13\) (wait, no, maybe \(AB\) is the hypotenuse? Wait, no, in the triangle, the side with length 13: let's use sine. \(\sin(30^{\circ})=\frac{x}{13}\), because \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\), and \(x\) is opposite the 30° angle, and 13 is the hypotenuse? Wait, no, in a right - triangle, the hypotenuse is the longest side. Wait, maybe I got the sides wrong. Wait, let's check again. If the triangle has a right angle and a 30° angle, then the side opposite 30° is half the hypotenuse. Wait, if 13 is the hypotenuse, then the side opposite 30° (which is \(x\)) would be \(\frac{13}{2}=6.5\)? Wait, no, \(\sin(30^{\circ})=\frac{1}{2}\), so \(\sin(30^{\circ})=\frac{x}{13}\), so \(x = 13\times\sin(30^{\circ})\).

Step2: Calculate \(x\)

We know that \(\sin(30^{\circ})=\frac{1}{2}\). So \(x = 13\times\frac{1}{2}=6.5\) or \(\frac{13}{2}\).

Answer:

\(x = \frac{13}{2}\) (or \(x = 6.5\))