In any triangle XYZ, the bisectors of the angles y and z meet at '0'. From a point 'O', OP perpendicular to YZ, OQ perpendicular to ZX AND OR perpendicular to XY respectively.Prove that OP=OQ=OR

 

Given: In any triangle XYZ, the bisectors of the angles y and z meet at O. 

From a point 'O', OP perpendicular to YZ, OQ perpendicular to ZX and  OR perpendicular to XY respectively. 

To Prove: OP=OQ=OR

Proof: 

In ΔYRO and ΔYPO,

∠RXO=∠PYO [because YO is the bisector of ∠Y]

YRO=∠YPO [ each equal to 90°]

YO = YO[common]

⇒OP = OR (cpct)...(i)

similarly, 

In  ΔZPO and ΔZQO, 

⇒OP = OQ (cpct)....(ii)

and In  

ΔXRO and ΔXQO, 

⇒OQ = OR (cpct).....(iii)

from (i), (ii) and (iii) ,we get,

OP = OQ = OR

Hence the result.

 

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