The vertices of a triangle lie on the circumference of a circle. If its sides are 2.5 cm, 6 cm, and 6.5 cm, what is the area of the circle (in c
m
2
)?

The area of the circle is approximately 33.183 cm²

Step-by-step explanation:

The sides of the inscribed triangle are 2.5 cm, 6 cm, and 6.5 cm

By cosine rule, we have;

6.5² = 6² + 2.5² - 2 × 6 × 2.5 × cos A

2 × 6 × 2.5 × cos A = 6² + 2.5² - 6.5² = 0

∴ cos(A) = 0

A = cos⁻¹(0) = 90°

The angle subtended at the center = 2 × Angle subtended at the circumference

Therefore, angle at the center = 2 × 90° = 180°

The 6.5 cm side forms an angle of 180° (a straight line) at the center of the circle and therefore, the 6.5 cm side = The diameter of the circle

The radius of the circle = 1/2 × The diameter of the circle = 1/2 × 6.5 cm = 3.25 cm

The area of the circle = π × r² = π × 3.25² ≈ 33.183 cm².