Tangents and secants

Geometry Level pending

$PA$ and $PB$ are tangents to circle $\beta$ and $PCD$ is a secant. Chords $AC,BC,BD$ and $DA$ are drawn. If $AC=9,AD=12$ and $BD=10$ , find $BC$ . If your answer can be expressed as $\dfrac{a}{b}$ where $a$ and $b$ are positive coprime integers, give your answer as $a+b$ .

The answer is 17.

1 solution

A Former Brilliant Member
Oct 25, 2017

Since $\angle PBC=\angle PDB$ , $\triangle DPB \sim \triangle BPC$ . Then

$\dfrac{CB}{DB}=\dfrac{PB}{PD}$

But $PA=PB$ , so

$\dfrac{CB}{DB}=\dfrac{PA}{PD}$ $\color{#D61F06}(1)$

Since $\angle PAC=PDA$ , $\triangle DAP \sim \triangle ACP$ . Then

$\dfrac{AC}{AD}=\dfrac{PA}{PD}$ $\color{#D61F06}(1)$

From $\color{#D61F06}(1)$ and $\color{#D61F06}(2)$ , we have

$\dfrac{AC}{AD}=\dfrac{CB}{DB}$

$\dfrac{9}{12}=\dfrac{CB}{10}$

$CB=\dfrac{90}{12}=\dfrac{15}{2}$

The desired answer is $15+2=$ $\boxed{17}$

Tangents and secants

The answer is 17.

1 solution

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