Question

find the value of x and y.
right triangle with right angle, vertical leg 12, horizontal leg y, hypotenuse x, and angle 30° at the end of horizontal leg

Explanation:

Step1: Find \( x \) using sine

In a 30 - 60 - 90 right triangle, the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is \( \sqrt{3} \) times the shorter leg. Here, the side with length 12 is opposite the 30° angle? Wait, no. Wait, the right angle, one angle is 30°, so the sides: the side opposite 30° is the shorter leg. Wait, the leg with length 12: let's check the angles. The right angle, 30°, so the third angle is 60°. So the side opposite 30° is the shorter leg, opposite 60° is the longer leg, and hypotenuse is \( x \). Wait, the leg of length 12: is it opposite 30° or 60°? Wait, the angle given is 30°, so the side opposite 30° is the shorter leg. Wait, the leg adjacent to 30° is \( y \), the leg opposite 30° is 12? Wait, no, the right angle is between the leg of length 12 and \( y \). So the angle of 30° is at the end of \( y \) and \( x \). So the side opposite 30° is 12, the hypotenuse is \( x \), and the adjacent side is \( y \).

In a right triangle, \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). So \( \sin(30^{\circ})=\frac{12}{x} \). Since \( \sin(30^{\circ}) = \frac{1}{2} \), we have \( \frac{1}{2}=\frac{12}{x} \), so \( x = 12\times2=24 \).

Step2: Find \( y \) using tangent or Pythagoras

We can use Pythagoras: \( a^{2}+b^{2}=c^{2} \), where \( a = 12 \), \( c=x = 24 \), so \( 12^{2}+y^{2}=24^{2} \). \( 144 + y^{2}=576 \), \( y^{2}=576 - 144=432 \), \( y=\sqrt{432}=\sqrt{144\times3}=12\sqrt{3} \).

Or using tangent: \( \tan(30^{\circ})=\frac{12}{y} \), but wait, no, \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Wait, \( \theta = 30^{\circ} \), opposite is 12, adjacent is \( y \), so \( \tan(30^{\circ})=\frac{12}{y} \), but \( \tan(30^{\circ})=\frac{1}{\sqrt{3}} \), so \( \frac{1}{\sqrt{3}}=\frac{12}{y} \), \( y = 12\sqrt{3} \), which matches the Pythagoras result.

Answer:

\( x = 24 \), \( y = 12\sqrt{3} \)