Question

Show that the length of tangents drawn from an external point to a circle are equal.

Answer 1

Answer:

Given:

Let P be an external point to a circle with centre, O and radius, r. Let PA and PB be the lengths of tangents from P to this circle.

To prove:

PA = PB.

Construction:

Join OA, OB, and OP.

Proof:

By tangent-radius theorem,

→ angle OAP = 90°.

→ angle OBP = 90°.

So,

∆OAP and ∆OBP are right angled triangles.

In ∆OAP and ∆OBP,

angle OAP = angle OBP = 90°.

OA = OB (radii of the same circle).

OP = OP (common).

=> ∆OAP ~= ∆OBP (by RHS congruency).

→ PA = PB (by CPCT).

Therefore,

The length of tangents drawn from an external point to a circle are equal.

Hence, proved.