Question

State and prove the tangent segment theorem

Answer 1

Step-1 Statement and construction: The segments of the tangent drawn from a point outside (external point) to a circle are congruent.

1. From an external point $D$ draw two tangents to the circle.
2. Let these tangents touch the circle at points $P,Q$
3. Join the center $A$ to the points $P,Q,D$.

Step-2 Proof of theorem:

Consider $△APD$ and $△AQD$

$segAD\cong segAD$ …(common side to both triangles)

$segAP\cong segAQ$ …(both radii of the circle)

A tangent is perpendicular to the radius at point of contact

$\Rightarrow$$AP\perp PD,AQ\perp QD$

$\Rightarrow$$\angle APD=\angle AQD=90°$

Thus by RHL congruency

$△APD\cong$$△AQD$

$\Rightarrow$$segPD\cong segQD$ …(By CPCT)

Hence, it is proved that the segments of the tangent drawn from a point outside (external point) to a circle are congruent