Question

PQR is a triangle with altitudes QM and RN to sides PR and PQ respectively are equal. Show that
∆PQM ≅∆PRN
PQ=PR

Answer 1

Given : PQR is a triangle with altitudes QM and RN to sides PR and PQ respectively are equal.

To Find : Show that

∆PQM ≅∆PRN

PQ=PR

Solution:

∆PQM  & ∆PRN

∠P = ∠P   common

∠PMQ = ∠PNR = 90°

QM = RN Given

Hence ∆PQM   ≅ ∆PRN

=> PQ = PR

Another way to show PQ = PR

Here if Triangle PQR  

= (1/2) * PQ * RN =  (1/2) * PR * QM

QM = RN

=> PQ = PR

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Answer 2

Answer:

Answer

We have,

According to given figure.

PQ=PR(giventhat)

QS=SR(Bydefinationofmidpoint)

PS=PS(Commonline)

Then,

ΔSPQ≅ΔSPR (BY congruency S.S.S.)

Hence, PS bisects ∠PQR by definition of angle bisector.