Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Find the answer

Find $PB$ given that $AP$ is a tangent to $\Omega$ , $\angle PAB=\angle PCA$ , and $\frac{PB}{PA}=\frac{4}{7}=\frac{PA}{PB+6}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since $AP$ is a tangent to $\Omega$ , we know that $\angle PAB=\angle PCA$ , so $\triangle PAB \sim \triangle PCA$ , so we get that $\frac{PB}{PA}=\frac{4}{7}=\frac{PA}{PB+6}$ Solving, we get that $7PB=4PA$ , so $4(PB+6)=7PA=\frac{49}{4}PB \Rightarrow \frac{33}{4}PB=24 \Rightarrow PB=\frac{32}{11}$

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