Question

in an isosceles triangle PQR where PQ=PR the bisectors of angle Q and R meet at I. Show that QI = RI and PI is the bisector of angle P.​

Answer 1

Given : an isosceles triangle PQR where PQ=PR

the bisectors of angle Q and R meet at I.

To find : Show that QI = RI and

PI is the bisector of angle P.​

Solution:

isosceles triangle PQR

PQ = PR

=> ∠Q  = ∠R   ( angles opposite to equal sides are equal )

QI and RI are bisector of angle Q & angle R

=> ∠RQI = ∠Q/2

   ∠QRI = ∠R/2

∠Q = ∠R  => ∠Q/2 = ∠R/2

=> ∠RQI =    ∠QRI

=>     RI  = QI ( Sides opposites to equal angles are equal )

∠PQI  = ∠Q/2

∠PRI  = ∠R/2

=> ∠PQI  = ∠PRI

in ΔPQI  and ΔPRI

PQ = PR    ( given )

∠PQI  = ∠PRI

QI  = RI     (shown above )

=>  ΔPQI  ≅ ΔPRI   (SAS)

=> ∠QPI = ∠RPI

=> PI is bisector of ∠P

Shown that QI = RI  and PI is bisector of ∠P

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