Question

find the area of the triangle using the following information. round to the nearest hundredth, if necessary.

a = 120°, b = 4, c = 14

Explanation:

Step1: Recall the formula for the area of a triangle

The formula for the area of a triangle when two sides \(b\), \(c\) and the included angle \(A\) are known is \(S=\frac{1}{2}bc\sin A\).

Step2: Substitute the given values into the formula

Given \(b = 4\), \(c=14\), and \(A = 120^{\circ}\). Since \(\sin120^{\circ}=\sin(180^{\circ} - 60^{\circ})=\sin60^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\).
Substitute into the formula: \(S=\frac{1}{2}\times4\times14\times\sin120^{\circ}\).
First, calculate \(\frac{1}{2}\times4\times14 = 28\).
Then, \(S = 28\times\sin120^{\circ}\).

Step3: Calculate the value of the area

\(S=28\times0.866 = 24.248\)

Answer:

\(24.25\)