# I need help answering this question.Point A is outside a circle and AB and AC are tangent to the circle at B and C, respectively. Point P and R are on AB and AC, respectively, PR is tangent to the...

I need help answering this question.

Point A is outside a circle and* AB *and *AC* are tangent to the circle at *B* and *C*, respectively. Point *P* and *R* are on *AB* and *AC*, respectively, *PR *is tangent to the circle at *Q*, and *AB*=20. Find the perimeter of triangle *APR.*

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### 1 Answer

We draw the figure.

Let A be any point out side a circle with centre O and radius r.

Since AB and AC are two external tangents to the circle at points B and C, the lengths of tangents are equal. So AB= AC. But AB = 20 given. Therefore AC = 20.

PQR is a tangent to the circle at Q. P is on AB and R is on AC. Therefore P is an external point to the circle . PB and PQ are two external tangents to the circle.

So PB = PQ ....(1)

Similarly RC = RQ......(2) as RC and RQ, as RC and RQ are tangets to the circle from an external point.

Add AP to both sides of eq(1):

AP+PB = AP+PQ. Or

AB = AP+PQ........(3).

Add AR to both sides of eq(2):

AR+RC = AR+RQ.

AC = AR+RQ............(4).

Add eq (3) and eq (4):

AB+AC = AP+PQ+AR+ RQ. Or

AB+AC = AP+PQ+QR +AR

AP+AC = AP+ PR+AR = Perimeter of the triangle APR.

20+20 = 40 Perimeter of triangle APR

Therefore the perimeter of triangle APR = 40.

Perimeter