In the adjoining figure, $$A$$ is the midpoint of the arc $$BAC$$. Given that $$AB = 15$$ and $$AD = 10$$, the value of $$AB$$ is
Solution
The uploaded paper prints $$AB$$ in the final sentence, but $$AB=15$$ is already given and the diagram and options correspond to $$AE$$. Let $$M$$ be the midpoint of $$BC$$. Since $$AB=AC$$, $$AM$$ is perpendicular to $$BC$$, and $$DB\cdot DC = AB^2-AD^2 = 225-100=125$$. By the intersecting chords theorem, $$AD\cdot DE=DB\cdot DC$$, so $$DE=12.5$$ and $$AE=AD+DE=22.5$$.
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