Question

prove that the tangent at the exteremites of any chord of a circle make equal angle with
the Chord​

Answer 1

Answer:

Given:AB is chord of circle with centre O.PA and PB are tangents at  extremities of any chord AB

To Prove :∠PAC=∠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,

∠CAP=∠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≅△PBC

∠PAC=∠PBC

Hence Proved