Question

If in \( \triangle ABC \), \( AB = 13 \, \text{cm} \), \( BC = 12 \, \text{cm} \), and the value of \( \angle C \) is:

• A. \( 90^\circ \)
• B. \( 30^\circ \)
• C. \( 60^\circ \)
• D. \( 45^\circ \)

Hint:

The cosine rule helps to find angles in a triangle when the lengths of all sides are known.

Answer 1

The Correct Option is C

We can apply the cosine rule to find the angle \( C \): \[ \cos C = \frac{AB^2 + BC^2 - AC^2}{2 \cdot AB \cdot BC}. \] We are given that \( AB = 13 \, \text{cm} \), \( BC = 12 \, \text{cm} \), and using the Pythagorean theorem, \( AC = \sqrt{13^2 - 12^2} = 5 \, \text{cm} \). Substituting the values: \[ \cos C = \frac{13^2 + 12^2 - 5^2}{2 \cdot 13 \cdot 12} = \frac{169 + 144 - 25}{2 \cdot 13 \cdot 12} = \frac{288}{312} = 0.923. \] Thus, \( \angle C = \cos^{-1}(0.923) = 60^\circ \). Thus, the correct answer is \( \boxed{60^\circ} \).