Question

The diagram shows a circle with centre O.

A, B & C lie on the circumference of this circle.
Given that AC is a diameter of the circle and ∠BCA = 5 × ∠BAC, find the size of ∠BCA as highlighted in the diagram.

Answer 1

∠ BCA = 75°

Explanation:

∠ ABC = 90° ( angle in a semicircle )

then the 2 remaining angles in Δ ABC = 90° , that is

∠ BAC + ∠ BCA = 90° ( ∠ BCA = 5 × ∠ BAC )

∠ BAC + 5 ∠ BAC = 90°

6 ∠ BAC = 90° ( divide both sides by 6 )

∠ BAC = 15°

Then

∠ BCA = 90° - 15° = 75°