Question

find y.
triangle image: right triangle with right angle, 60° angle, 30° angle, one leg is 11√3 yd, the other leg is y
write your answer in simplest radical form.
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Explanation:

Step1: Identify triangle type

This is a 30-60-90 right triangle. In such a triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\) (opposite 30°, 60°, 90° respectively).

Step2: Relate sides to angles

The side opposite 30° is \(y\)? Wait, no—wait, the right angle, 60°, 30°: the side adjacent to 60° (and opposite 30°) is \(y\)? Wait, no, let's label: right angle, 60°, 30°. So the side opposite 30° is \(y\)? Wait, no, the given side is \(11\sqrt{3}\) yd, which is adjacent to 30° (since it's the leg with the right angle and 30° angle). Wait, in a 30-60-90 triangle, \(\tan(30^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{y}{11\sqrt{3}}\)? Wait, no, \(\tan(60^\circ)=\frac{11\sqrt{3}}{y}\)? Wait, let's clarify angles. The right angle, one angle 60°, one 30°. So the side opposite 60° is \(11\sqrt{3}\), and the side opposite 30° is \(y\). In a 30-60-90 triangle, \(\tan(60^\circ)=\sqrt{3}=\frac{\text{opposite 60°}}{\text{opposite 30°}}=\frac{11\sqrt{3}}{y}\).

Step3: Solve for \(y\)

From \(\sqrt{3}=\frac{11\sqrt{3}}{y}\), multiply both sides by \(y\): \(y\sqrt{3}=11\sqrt{3}\). Divide both sides by \(\sqrt{3}\): \(y = 11\)? Wait, no, that can't be. Wait, maybe I mixed up. Wait, in 30-60-90, the sides are: let the side opposite 30° be \(x\), then opposite 60° is \(x\sqrt{3}\), hypotenuse \(2x\). So here, the side opposite 60° is \(11\sqrt{3}\), so \(x\sqrt{3}=11\sqrt{3}\), so \(x = 11\). But the side opposite 30° is \(x\), which is \(y\)? Wait, the right angle, 60°, 30°: so the legs are \(y\) (opposite 30°) and \(11\sqrt{3}\) (opposite 60°), hypotenuse. So yes, in 30-60-90, leg opposite 30° is \(x\), leg opposite 60° is \(x\sqrt{3}\). So \(x\sqrt{3}=11\sqrt{3}\) ⇒ \(x = 11\). So \(y = 11\)? Wait, but let's check with tangent. \(\tan(30^\circ)=\frac{y}{11\sqrt{3}}=\frac{1}{\sqrt{3}}\). So \(y=\frac{11\sqrt{3}}{\sqrt{3}} = 11\). Yes, that works.

Answer:

\(11\)