Question

The figure shows a circle of diameter AB and radius 6.5 cm. If chord CA is 5 cm long, find the area of triangle ∆ABC.

Hint:

If a triangle is inscribed in a circle with one side as diameter, it's a right triangle.

Answer 1

Correct Answer: 30

Step 1: Since AB is the diameter of the circle, triangle ∆ABC is a right triangle with ∠C = 90°.
Step 2: Use Pythagoras theorem in triangle ABC to find BC.
AB = 13 cm (diameter), CA = 5 cm
Let BC = x. By Pythagoras theorem: \[ AB^2 = AC^2 + BC^2 \Rightarrow 13^2 = 5^2 + x^2 \Rightarrow 169 = 25 + x^2 \Rightarrow x^2 = 144 \Rightarrow x = 12 \text{ cm} \] Step 3: Use the formula for the area of a right-angled triangle: \[ \text{Area} = \frac{1}{2} \times AC \times BC = \frac{1}{2} \times 5 \times 12 = 30 \text{ cm}^2 \] Note: There is a discrepancy! Since radius = 6.5 cm, diameter = 13 cm, and triangle is right-angled at C, verify side BC: \[ AC = 5, AB = 13 \Rightarrow BC = \sqrt{13^2 - 5^2} = \sqrt{144} = 12 \text{ cm} \] Area = ½ × AC × BC = ½ × 5 × 12 = 30 cm².