# B is 34 mm from the center of circle O, which has a radius of 16 mm. Line BP and line BR are tangent segments. Line AC is tangent to circle O at point Q. Find the perimeter of ABC?

Oct 25, 2017

perimeter $= 60$ mm

#### Explanation:

$B R = \sqrt{B {O}^{2} - O {R}^{2}} = \sqrt{{34}^{2} - {16}^{2}} = \sqrt{900} = 30$
as the two tangent segments to a circle from an external point are equal in length, $\implies \textcolor{red}{B R = B P , A P = A Q , \mathmr{and} C R = C Q}$
perimeter of $\Delta A B C = A B + B C + A C$
$= A B + B C + A Q + Q C$
$= \left(A B + A Q\right) + \left(B C + Q C\right)$
$\implies \left(A B + A P\right) + \left(B C + C R\right)$
$\implies B P + B R$
$= 30 + 30 = 60$ mm