CEMC Fermat, 2024

Geometry Difficulty 1.3 Junior Find the answer Canada

In the diagram, a semi-circle has centre $O$ and diameter $AC$ . Also, $B$ is a point on the circumference such
that $\angle BAC = 25\degree$ .

The measure of $\angle BOC$ is

Pick one

Solution

Since $O$ is the centre of the
circle, then $OA = OB = OC$ .

This means that $\triangle AOB$ and
$\triangle COB$ are both isosceles
with $\angle ABO = \angle BAO = \angle BAC = 25\degree$ .

Thus, $\angle AOB = 180\degree - \angle ABO - \angle BAO = 130\degree$ .

Since $\angle AOC$ is a straight
angle, then $\angle BOC = 180\degree - \angle AOB = 180\degree - 130\degree = 50\degree$ .

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