Answer :

Consider the following figure

A circle with center O and radius r and A be the external point from which tangents AX and AY are drawn

A line drawn from center to the point of contact of tangent and circle is perpendicular to tangent

Therefore OX is perpendicular to AX and OY is perpendicular to AY

To prove: AX = AY

OX and OY are radius of circle

∴ OX = OY = r

Consider ∆AXO

∠AXO = 90°

∴ by Pythagoras

AX^{2} + XO^{2} = AO^{2}

AX^{2} = AO^{2} - XO^{2}

AX^{2} = AO^{2} - r^{2}

Consider ∆AYO

∠AYO = 90°

∴ by Pythagoras

AY^{2} + YO^{2} = AO^{2}

AY^{2} = AO^{2} - YO^{2}

AY^{2} = AO^{2} - r^{2}

From (i) and (ii) we conclude that

AX = AY

Hence proved

Therefore, the lengths of the two tangent segments to a circle drawn from an external point are equal.

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