Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.1 AMC 10/12 Find the answer

Two tangents are drawn to a circle from an exterior point $A$ ; they touch the circle at points $B$ and $C$ respectively.
A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$ , and touches the circle at $Q$ . If $AB=20$ , then the perimeter of $\triangle APR$ is

Pick one

Solution

1961 AHSME Problem 11.png
Draw the diagram as shown. Note that the two tangent lines from a single outside point of a circle have the exact same length, so $AB = AC = 20$ , $BP = PQ$ , and $QR = CR$ .
The perimeter of the triangle is $AP + PR + AR$ . Note that $PQ + QR = PR$ , so from substitution, the perimeter is
$AP + PQ + QR + AR$
$AP + BP + CR + AR$
$AB + AC$
Thus, the perimeter of the triangle is $40$ , so the answer is $\boxed{\textbf{(C)}}$ .

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