Solve any two sub-question:

Prove that, “The lengths of the two tangent segments to a circle drawn from an external point are equal.”

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

AX² + XO² = AO²

AX² = AO² - XO²

AX² = AO² - r²

Consider ∆AYO

∠AYO = 90°

∴ by Pythagoras

AY² + YO² = AO²

AY² = AO² - YO²

AY² = AO² - r²

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.