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# Practice Paper 1 Question 3

Triangle $$ABC$$ is isosceles with $$AB=AC$$. Let the circle having diameter $$AB$$ and centre $$O$$ intersect $$BC$$ at some point $$P$$. Find the ratio $$\frac{BP}{BC}$$.

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## Hints

• Hint 1
What can you say about the $$\angle APB$$?
• Hint 2
What about $$AP$$?

## Solution

Angle $$\angle APB=\pi/2$$ because it subscribes an arc length of $$\pi$$ (AB is a diameter), hence $$AP$$ is the height in the triangle. Being an isosceles triangle, this is also the median, and hence $$\frac{BP}{BC}=\frac{1}{2}$$.

Can you find an alternative proof considering the segment $$OP$$ instead?

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