Question

Seg PM and seg PN are congruent chords of a circle with centre C. Show that the ray PC is the bisector of ∠NPM.

Given: Point C is the centre of the circle. chord PM ≅ chord PN

To prove: Ray PC is the bisector of ∠NPM.

Construction: Draw seg CR ⊥ chord PN, P-R-N, seg CQ ⊥ chord PM, P-Q-M

Answer 1

chord PM chord PN [Given] 

seg CR ⊥ chord PN 

seg CQ ⊥ chord PM [Construction] 

∴ segCR ≅ segCQ ….(i) [Congruent chords are equidistant from the centre] 

In ∆PRC and ∆PQC, 

∠PRC ≅ ∠PQC [Each is of 90°] 

segCR ≅ segCQ [From (i)]

seg PC ≅ seg PC [Common side]

∴ ∆PRC ≅ ∆PQC [Hypotenuse side test] 

∴ ∠RPC ≅ ∠QPC [c. a. c. t.] 

∴ ∠NPC ≅ ∠MPC [N- R-P, M-Q-P] 

∴ Ray PC is the bisector of ∠NPM.