Question

1. determine the area of the following.

a)
(right triangle with angles 30°, 60°, right angle, side 12 opposite 60°? or adjacent? x, y labels, and √3 : 2 written)

Explanation:

Step1: Identify triangle type

This is a 30 - 60 - 90 right - triangle. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest side (let's call it \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\). Here, the hypotenuse is 12, so we can find the other two sides.
We know that hypotenuse \( = 2x=12\), so \(x = \frac{12}{2}=6\) (this is the side opposite \(30^{\circ}\)). The side opposite \(60^{\circ}\) (let's call it \(y\)) is \(x\sqrt{3}=6\sqrt{3}\).

Step2: Calculate area of triangle

The formula for the area of a right - triangle is \(A=\frac{1}{2}\times base\times height\). In a right - triangle, the two legs are the base and the height. Here, the legs are \(x = 6\) and \(y = 6\sqrt{3}\).
So, \(A=\frac{1}{2}\times6\times6\sqrt{3}\)
First, calculate \(6\times6\sqrt{3}=36\sqrt{3}\), then \(\frac{1}{2}\times36\sqrt{3}=18\sqrt{3}\)

Answer:

The area of the triangle is \(18\sqrt{3}\)