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Two distinct points $A$ and $B$ are on a circle with center at $O$, and point $P$ is outside the circle such that $\overline{PA}$ and $\overline{PB}$ are tangent to the circle. Find $AB$ if $PA = 12$ and the radius of the circle is 9.

AdminMod2 Sep 2, 2017

#1**0 **

Two distinct points $A$ and $B$ are on a circle with center at $O$, and point $P$ is outside the circle such that $\overline{PA}$ and $\overline{PB}$ are tangent to the circle. Find $AB$ if $PA = 12$ and the radius of the circle is 9.

Right triangle OPA

\(opposite\ side=9\\ adjacent\ side=12\\ tan\ \alpha=\frac{9}{12}\\ \color{blue}\alpha=atan\frac{9}{12}\)

Right triangle PAC

\(hypotenuse=12\\ opposite\ side=\frac{\overline{AB}}{2}\\ \alpha=atan\frac{9}{12}=atan\ 0.75\\ sin\alpha =\frac{\overline{AB}}{2}/12\\\color{blue} \overline{AB}=2\times 12\times sin \alpha=2\times 12\times sin(atan\ 0.75)\)

\(\overline {AB}=14.4\)

!

asinus Sep 2, 2017