Prove that the tangents at the extremities of any chord make equal angles with the chord. [3 MARKS]
Concept: 1 Mark
Application: 2 Marks
Let AB be a chord of a circle with center O, and let AP and BP be the tangents at A and B respectively.
Suppose the tangents meet at P .
Join OP. Suppose OP meets AB at C . We have to prove that ∠PAC=∠PBC .
In triangles PCA and PCB , We have
PA = PB
[The tangents are equally inclined to line joining extrernal point and centre of circle.]
PC = PC [Common]
So, by SAS - criterion of congruence, we have
In ΔPAB, PA=PB ( ∵ tangents drawn from an external point to a circle are equal.)
∵ PA = PB, ∠PAB = ∠ PBA (Equal opposite sides make equal angles)
⇒ ∠PAC = ∠PBC