In any triangle XYZ, the bisectors of the angles y and z meet at '0' From a point 'O', - Maths - Triangles

Question

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

Gursheen kaur. · Accepted Answer

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]

$t h e r e f o r e, Δ Y R O ≅ Δ Y P O [b y A A S c o n g r u e n c y c r i t e r i o n]$

⇒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.

9

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