Question

a right triangle has a 30° angle. the leg adjacent to the 30° angle measures 25 inches. what is the length of the other leg? round to the nearest tenth. 14.4 in. 43.3 in. 28.9 in. 21.7 in.

Explanation:

Step1: Identify trigonometric ratio

We know that in a right triangle, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 30^\circ\), adjacent side \(= 25\) in, and we need to find the opposite side (other leg). So \(\tan(30^\circ)=\frac{\text{opposite}}{25}\).

Step2: Solve for opposite side

We know that \(\tan(30^\circ)=\frac{1}{\sqrt{3}}\approx0.577\). So, \(\text{opposite}=25\times\tan(30^\circ)=25\times\frac{1}{\sqrt{3}}\approx25\times0.577 = 14.425\approx14.4\) in. Wait, no, wait. Wait, maybe I mixed up. Wait, if the angle is \(30^\circ\), and the adjacent is 25, maybe it's the other way? Wait, no, \(\tan(30^\circ)=\frac{\text{opposite}}{\text{adjacent}}\), so opposite \(=\) adjacent \(\times\tan(30^\circ)\). Wait, but maybe I made a mistake. Wait, no, let's recalculate. \(\tan(30^\circ)=\frac{1}{\sqrt{3}}\approx0.577\), so \(25\times0.577\approx14.4\). But wait, maybe it's \(\cot\)? No, \(\tan\) is opposite over adjacent. Wait, maybe the angle is \(30^\circ\), and the adjacent is 25, so the other leg (opposite) is \(25\times\tan(30^\circ)\approx14.4\). But let's check the options. One of the options is 14.4 in. Wait, but let's confirm. Alternatively, maybe it's a 30-60-90 triangle? Wait, in a 30-60-90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\). Wait, if the adjacent to 30° is the longer leg? Wait, no. Wait, in a 30-60-90 triangle, the side opposite 30° is the shortest leg, opposite 60° is the longer leg, and hypotenuse is twice the shortest leg. Wait, maybe I misidentified the angle. Wait, if the angle is 30°, and the adjacent side is 25, then if 30° is one of the acute angles, then the side opposite 30° is the shorter leg, and adjacent is the longer leg? Wait, no. Wait, in a right triangle, for angle \(\theta\), adjacent is adjacent to \(\theta\), opposite is opposite to \(\theta\). So if \(\theta = 30^\circ\), adjacent is 25, then opposite is \(25\times\tan(30^\circ)\approx14.4\), which matches one of the options (14.4 in). Wait, but let's recalculate \(\tan(30^\circ)\): \(\tan(30^\circ)=\frac{\sqrt{3}}{3}\approx0.577\), so \(25\times0.577 = 14.425\approx14.4\) in. So the other leg is approximately 14.4 in.

Answer:

14.4 in.