Question

options: ( x = \frac{10sqrt{3}}{3}, y = \frac{5sqrt{3}}{3} ); ( x = 10sqrt{3}, y = \frac{5sqrt{3}}{3} ); ( x = 10, y = 5sqrt{3} ); ( x = \frac{5sqrt{3}}{3}, y = 10 )

Explanation:

Step1: Identify triangle type

This is a right - triangle with one angle \(60^{\circ}\), so the other non - right angle is \(30^{\circ}\). In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (opposite to \(30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\) respectively). The side of length \(5\) is opposite the \(30^{\circ}\) angle.

Step2: Find hypotenuse \(x\)

In a \(30 - 60 - 90\) triangle, the hypotenuse is twice the length of the side opposite the \(30^{\circ}\) angle. Let the side opposite \(30^{\circ}\) be \(a = 5\), then hypotenuse \(x=2a\). So \(x = 2\times5=10\).

Step3: Find side \(y\) (opposite \(60^{\circ}\))

The side opposite \(60^{\circ}\) is \(\sqrt{3}\) times the side opposite \(30^{\circ}\). So \(y = 5\sqrt{3}\) (since the side opposite \(30^{\circ}\) is \(5\)).

Answer:

\(x = 10,y = 5\sqrt{3}\) (the option with \(x = 10,y = 5\sqrt{3}\))