Question

Prove any theorem of chapter Circles. ​

Answer 1

Theorem 1

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Construction: Draw a circle with centre O. Draw a tangent XY which touches point P at the circle.

To Prove: OP is perpendicular to XY.

Draw a point Q on XY; other than O and join OQ. Here OQ is longer than the radius OP.

OQ > OP

For every point on the line XY other than O, like Q1, Q2, Q3, ……….Qn;

OQ1>OP

OQ2>OP

OQ3>OP

OQ4>OP

Since OP is the shortest line

Hence, OP ⊥ XY proved

Theorem 2

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

Construction: Draw a circle with centre O. From a point P outside the circle, draw two tangents P and R.

10 circle theorem 1

To Prove: PQ = PR

Proof: In Δ POQ and Δ POR

OQ=OR (radii)

PO=PO(common side)

∠PQO=∠PRO (Right angle)

Hence;

ΔPOQ≅ΔPOR proved