Question

find the measure of angle a.
a. 60°
b. 135°
c. 120°
d. 140°

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(BC^{2}=AB^{2}+AC^{2}-2(AB)(AC)\cos A\).
Given \(AB = 2\), \(AC=6\), \(BC = 2\sqrt{13}\).
Substitute into the formula: \((2\sqrt{13})^{2}=2^{2}+6^{2}-2\times2\times6\times\cos A\).
Simplify: \(52 = 4 + 36-24\cos A\).

Step2: Solve for \(\cos A\)

First, calculate \(4 + 36=40\).
The equation becomes \(52=40 - 24\cos A\).
Rearrange: \(24\cos A=40 - 52\).
\(24\cos A=-12\).
Then \(\cos A=\frac{-12}{24}=-\frac{1}{2}\).

Step3: Find the angle \(A\)

Since \(A\in(0,\pi)\) and \(\cos A =-\frac{1}{2}\), then \(A = 120^{\circ}\).

Answer:

C. \(120^{\circ}\)