Question

Prove that the tangents at the extremities of any chord make equal angles with the chord.

Answer 1

Given:AB is chord of circle with centre O.PA and PB are tangents at extremities of any chord AB
To Prove :$\angle$PAC$=$$\angle$PBC
Proof:
Let AB be a chord of a circle with centre O, and let AP and BP be the tangents at A and B respectively.
Suppose, the tangents meet at point P. Join OP.
Suppose OP meets AB at C.
In triangles $△$PCA and$△$ PCB,
$\angle$CAP$=$$\angle$CBP(Line joining point of contact to center is perpendicular to tangent)
PA$=$PB [PA and PB are equally inclined to OP]
And PC$=$PC [Common]
So, by SAS criteria of congruence
$△$PAC$\cong$$△$PBC
$\angle$PAC$=\angle$PBC
Hence Proved