Prove that the tangents at the exterimites Of any chord makes equal an

1.	Prove that the tangents at the exterimites Of any chord makes equal angles with the chord.
Answer» \xa0Let AB be a chord of a circle with centre O, and let AP and BP be the tangents at A and B respectively. Let the two tangents AP and BP meets at P.\xa0Now Join OP. Suppose OP meets AB at C and the chord AB=AC+BC.We have to prove that\xa0{tex}\\angle P A C = \\angle P B C{/tex}In two triangles PCA and PCB, we havePA = PB\xa0[{tex} \\because{/tex}Tangents from an external point are equal]{tex}\\angle A P C = \\angle B P C{/tex}\xa0[{tex} \\because {/tex}\xa0PA and\xa0PB are equally inclined to OP]and, PC = PC [Common]So, by SAS-criterion of congruence, we obtain{tex}\\Delta P A C \\cong \\Delta P B C{/tex}{tex}\\Rightarrow \\quad \\angle P A C = \\angle P B C{/tex}

Discussion