Question

The vertex angle of an isosceles triangle is three times the measure of either base angle. Find the area if the measure of the congruent sides are 14 in

Answer 1

`Area = 93.21in^2`

Explanation:

Given

`\theta \to` vertex angle

`\alpha \to` base angle

`\theta = 3\alpha`

`l = 14in` -- congruent sides

Required

The area of the triangle

First, calculate the angles

`\theta + \alpha + \alpha = 180^o` ----- angles in an isosceles triangle

Substitute
`\theta = 3\alpha`

`3\alpha + \alpha + \alpha = 180^o`

`5\alpha = 180^o`

Divide both sides by 5

`\alpha = 36^o`

Recall that:
`\theta = 3\alpha`

`\theta = 3 * 36^o`

`\theta = 108^o`

The area is then calculated as:

`Area = (1)/(2)l^2 \sin(\theta)`

`Area = (1)/(2)*14^2 \sin(108^o)`

`Area = (1)/(2)*196 *0.9511`

`Area = 93.21in^2`