Question

find the values of x and y. write your answers in simplest form.
x =
y =
(there is a right triangle with one angle 60° and the side adjacent to 60° is 3√3, x is the side opposite to 60°, y is the hypotenuse)

Explanation:

Step1: Identify Triangle Type

The triangle is a right - triangle with one angle \(60^{\circ}\), so it's a \(30 - 60 - 90\) triangle. In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (opposite \(30^{\circ}\), \(60^{\circ}\), and hypotenuse respectively). The side opposite \(60^{\circ}\) is \(3\sqrt{3}\), let the side opposite \(30^{\circ}\) be \(x\) and hypotenuse be \(y\).

Step2: Find \(x\) (opposite \(30^{\circ}\))

We know that \(\tan(60^{\circ})=\frac{\text{opposite}}{\text{adjacent}}=\frac{3\sqrt{3}}{x}\). Since \(\tan(60^{\circ}) = \sqrt{3}\), we have \(\sqrt{3}=\frac{3\sqrt{3}}{x}\). Cross - multiplying gives \(x\sqrt{3}=3\sqrt{3}\), so \(x = 3\).

Step3: Find \(y\) (hypotenuse)

Using the Pythagorean theorem \(y^{2}=x^{2}+(3\sqrt{3})^{2}\). Substitute \(x = 3\): \(y^{2}=3^{2}+(3\sqrt{3})^{2}=9 + 27=36\), so \(y = 6\). Or using the ratio in \(30 - 60 - 90\) triangle, hypotenuse is twice the side opposite \(30^{\circ}\), so \(y = 2x=2\times3 = 6\).

Answer:

\(x = 3\), \(y = 6\)