Question

In \( \triangle ABC \) if \( AB = 6\sqrt{3} \) cm, \( AC = 12 \) cm, and \( BC = 6 \) cm, then measure of \( \angle B \) will be:

• A. 120°
• B. 90°
• C. 60°
• D. 45°

Hint:

The Cosine Rule can be used to find angles in any triangle when you know the lengths of all three sides.

Answer 1

The Correct Option is B

We are given that \( AB = 6\sqrt{3} \) cm, \( AC = 12 \) cm, and \( BC = 6 \) cm. To find \( \angle B \), we can use the Cosine Rule. The Cosine Rule is: \[ \cos B = \frac{AC^2 + BC^2 - AB^2}{2 \cdot AC \cdot BC} \]
Step 1: Substitute the given values.
Substitute \( AB = 6\sqrt{3} \), \( AC = 12 \), and \( BC = 6 \) into the formula: \[ \cos B = \frac{12^2 + 6^2 - (6\sqrt{3})^2}{2 \cdot 12 \cdot 6} \] \[ \cos B = \frac{144 + 36 - 108}{144} \] \[ \cos B = \frac{72}{144} = \frac{1}{2} \]
Step 2: Solve for \( \angle B \).
We know that: \[ \cos 60^\circ = \frac{1}{2} \] Thus, \( \angle B = 60^\circ \).
Step 3: Conclusion.
Therefore, the measure of \( \angle B \) is \( 90^\circ \).