EXERCISE 10.1

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, ……….Q_n;

$O{Q}_1>OP$

$O{Q}_2>OP$

$O{Q}_3>OP$

$O{Q}_4>OP$

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.

To Prove: PQ = PR

Proof: In Δ POQ and Δ POR

$OQ=OR$ (radii)

$PO=PO$ (common side)

$\angle PQO=\angle PRO$ (Right angle)

Hence; $\Delta POQ\cong \Delta POR$ proved