Question

in triangle PQR, angle QPR is bisected by PS and PT perpendicular to RQ,then show that angle TPS =1/2(angle R-angle Q)

Answer 1

Given; PS is bisector of angle QPR

PT is perpendicular to QR

To Prove; angle TPS = 1/2 (angle Q - angle R)

Solutio; We know that angle QPS = 1/2 angle P

(1) angle Q + angle QPT = 90 degree

angle QPT = 90 degree -angle Q (2) On putting equations (1) and (2)

angle TPS = angle QPS - angle QPT

= 1/2 angle P - ( 90 - angle Q )

= 1/2 angle P - 90 + angle Q

= 1/2 angle P - 1/2 (angle Q + angle P + angle R) + angle Q

= 1/2 angle P - 1/2 angle Q - 1/2 angle P - 1/2 angle R + angle Q

= -1/2 angle Q - 1/2 angle R + angle Q (+ve and -ve 1/2 angle P got cancled )

by takig LCM in angle Q we get

-1/2 angle Q + 2/2 angle Q - 1/2 angle R

(-1/2 + 2/2) angle Q - 1/2 angle R

= 1/2 angle Q - 1/2 angle R

angle TPS = 1/2 ( angle Q -1/2 angle R ) (Hence proved)

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