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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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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.
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