Prove that 'tangent segments drawn from an external point to a circle are congruent.
Answers
given's
→ PD and are two tangents. +. Dis a external points from where tangents. are drawn on circle.
To prove: Tangents to and ces are congruent.
Answer:
Given: A circle with centre O; PA and PB are two tangents to the circle drawn from an external point P.
To prove: PA = PB
Construction: Join OA, OB, and OP.
It is known that a tangent at any point of a circle is perpendicular to the radius through the point of contact.
OA⊥PAOB⊥PB
In triangle OPA and OPB
∠OPA=∠OPBOA=OB(radii)OP=OP(common)
Therefore triangle OPA is congruent to triangle OPB by RHS criterion.
Which means PA=PB
(Corresponding parts of congruent triangles are equal)
Thus, it is proved that the lengths of the two tangents drawn from an external point to a circle are equal.
The length of tangents drawn from any external point are equal.