Question

State the following statement is True or False

The length of two tangents drawn from an external point to a circle are not equal.

• A. True
• B. False

Answer 1

The correct option is B False

We are given that the length of tangents drawn from any external point are not equal. But we will prove that the length of tangents drawn from any external point are equal.

Proof is as follows:

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.

$OA\perp PA$ $-\left(1\right)$

$OB\perp PB$ $-\left(2\right)$

In $△OPA$ and $△OPB$

$\angle OPA=\angle OPB$ (Using $\left(1\right)$ and $\left(2\right)$ they are equal to ${90}^o$)

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

$OP=OP$ (Common side)

Therefor $△OPA\cong △OPB$(RHS congruency criterion)

$PA=PB$

(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.

So the asked statement is incorrect.