Question

find x and y using special right triangles. write your answer in simplified radical form. 3 30° x y 60°

Explanation:

Step1: Use the property of 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\). The side opposite the 30° angle is the shortest side. Let the side opposite 30° be \(a\), the side opposite 60° be \(a\sqrt{3}\), and the hypotenuse be \(2a\). Here, the side opposite 30° is \(x\), the side opposite 60° is 3, and the hypotenuse is \(y\).
Since the side opposite 60° is \(a\sqrt{3}\) and it is equal to 3. So, if \(a\sqrt{3}=3\), then \(a = \frac{3}{\sqrt{3}}=\sqrt{3}\).

Step2: Find \(x\)

\(x\) is the side opposite 30°, so \(x = a=\sqrt{3}\)

Step3: Find \(y\)

\(y\) is the hypotenuse. Since \(y = 2a\) and \(a=\sqrt{3}\), then \(y = 2\sqrt{3}\)

Answer:

\(x=\sqrt{3}\), \(y = 2\sqrt{3}\)