Question

the angle measures in a triangle are in the ratio 1:3:5. the longest side of the triangle measures 5 inches. what is the approximate perimeter of the triangle? 9.0 in 9.4 in 11.1 in 12.7 in

Explanation:

Step1: Find the angles of the triangle

Let the angles be \(x\), \(3x\), and \(5x\). Since the sum of angles in a triangle is \(180^{\circ}\), we have \(x + 3x+5x=180^{\circ}\).

$$9x = 180^{\circ}$$

$$x = 20^{\circ}$$

So the angles are \(20^{\circ}\), \(60^{\circ}\), and \(100^{\circ}\).

Step2: Use the Law of Sines

Let the sides opposite to the angles \(20^{\circ}\), \(60^{\circ}\), and \(100^{\circ}\) be \(a\), \(b\), and \(c = 5\) (longest side opposite the largest angle \(100^{\circ}\)).
By the Law of Sines \(\frac{a}{\sin20^{\circ}}=\frac{b}{\sin60^{\circ}}=\frac{c}{\sin100^{\circ}}\).
First, find \(a\):

$$a=\frac{c\sin20^{\circ}}{\sin100^{\circ}}=\frac{5\times\sin20^{\circ}}{\sin100^{\circ}}\approx\frac{5\times0.3420}{0.9848}\approx1.73$$

Then, find \(b\):

$$b=\frac{c\sin60^{\circ}}{\sin100^{\circ}}=\frac{5\times\sin60^{\circ}}{\sin100^{\circ}}\approx\frac{5\times0.8660}{0.9848}\approx4.4$$

Step3: Calculate the perimeter

Perimeter \(P=a + b + c\approx1.73+4.4 + 5=11.13\approx11.1\)

Answer:

11.1 in