Question

A is a point at a distance 13 cm from the center O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ΔABC.

Answer 1

Given : Two tangents are drawn from an external point A to the circle with center O, Tangent BC is drawn at a point R.

Radius of circle equals 5cm and OA = 13 cm

OA = 13 cm

To Find : Perimeter of ΔABC.

∠OPA = 90°

[tangent at any point of a circle is perpendicular to the radius through the point of contact]

(OA)² = (OP)² + (PA)²

(13)² = (5)² + (PA)²

169 - 25 = (PA)²

(PA)²= 144

PA = 12 cm

As we know that, Tangents drawn from an external point to a circle are equal.

So we have

PB = BR [1] [Tangents from point B]

CR = QC [2] [Tangents from point C]

Now Perimeter of Triangle PCD

= AB + BC + CA

= AB + BR + CR + CA

= AB + PB + QC + CA [From 1 and 2]

= PA + QA

Now,

PA = QA = 12 cm as tangents drawn from an external point to a circle are equal

So we have

Perimeter = PA + QA = 12 + 12 = 24 cm