Track / Stage 5 / 51 of 400 #651 of 2000

Problem 651

AIME late

Geometry Difficulty 5.2 Find the answer

6. A circle passes through vertices $A$ and $B$ of triangle $ABC$ and intersects its sides $AC$ and $BC$ at points $Q$ and $N$ respectively, such that $AQ: QC = 5: 2$ and $CN: NB = 5: 2$ . Find $AB$ , if $QN = 5 \sqrt{2}$ .

Your answer

Official solution

Answer: $7 \sqrt{5}$ .

Solution. Let $C Q=2 x, C N=5 y$ ; then $A Q=5 x, C T=2 y$ . By the theorem of two secants $C Q \cdot C A=C N \cdot C B$ , from which $2 x \cdot 7 x=5 y \cdot 7 y, y=x \sqrt{\frac{2}{5}}$ . Triangles $A B C$ and $N Q C$ are similar by two sides and the angle between them ( $C A: C N=C B: C Q, \quad \angle C-$ is common), and the similarity coefficient is $\frac{B C}{B Q}=\frac{7 y}{2 x}=\frac{7}{2} \sqrt{\frac{2}{5}}=\frac{7}{\sqrt{10}}$ . Therefore, $A B=Q N \cdot \frac{7}{\sqrt{10}}=7 \sqrt{5}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.