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Philippine Mathematical Olympiad

Geometry Difficulty 5.0 AIME Prove it Philippines

Problem:

The segments $A P$ and $A Q$ are tangent to circle $O$ at points $P$ and $Q$ , respectively. Moreover, $Q E$ is perpendicular to diameter $P D$ of length $4$ . If $P E = 3.6$ and $A P = 6$ , what is the length of $Q E$ ?

Solution

Solution:

Draw $A O$ , $P Q$ and $O Q$ . We note that $A O \perp P Q$ since $A Q = A P$ and $O Q = O P$ .

This implies that $\angle P A O \cong \angle E P Q$ (both complementary to $\angle A P R$ ).

Hence $\triangle P A O \approx \triangle E P Q$ . Therefore
$\frac{A P}{O P} = \frac{P E}{Q E} \Longrightarrow Q E = \frac{P E \times O P}{A P} \Longrightarrow Q E = \frac{3.6 \times 2}{6} = 1.2$

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