Question

Prove that the length of the tangent drawn from an external point

Answer 1

Step-by-step explanation:

Given: A circle with centre O; PA and PB are two tangents to the circle drawn from an external point P.

To prove: PA = PB

Construction: Join OA, OB, and OP.

It is known that a tangent at any point of a circle is perpendicular to the radius through the point of contact.

OA\bot PAOA⊥PA

OB\bot PBOB⊥PB

In \triangle OPA△OPA and \triangle OPB△OPB

\angle OPA=\angle OPB∠OPA=∠OPB (Using (1))

OA = OBOA=OB (Radii of the same circle)

OP = OPOP=OP (Common side)

Therefor \triangle OPA\cong \triangle OPB△OPA≅△OPB (RHS congruency criterion)

PA = PBPA=PB

(Corresponding parts of congruent triangles are equal)

Thus, it is proved that the lengths of the two tangents drawn from an external point to a circle are equal.

The length of tangents drawn from any external point are equal.

So statement is correct..

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