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Question : If ∆ PQR is an isosceles triangle, right angled at P and M is the point on QR such that, PM⊥QR. Show that PM²=QM.MR
[1 Mark Question] 

Doubt by Charu

Solution : 

Given : ∠ P = 90°
PM ⊥ QR
PQ=PR

To Prove : PM²=QM.MR

Proof : 
PR=PR (Given)
∠Q=∠R
[Angle opposite to equal sides of a triangle are equal]

In  ∆PMQ and  ∆PMR
∠Q=∠R [Proved above]

∠PMQ = ∠PMR [each 90°]

∆PMQ ~ ∆PMR [By AA similarity criteria]
PM/
We can do this question by three different methods.

Method 1) Similar the two triangles and by CPST derive the required result.

Method 2) Using Pythagoras Theorem 

Method 3) Similar the two triangles then apply area theorem and then apply the basic formula for area of triangle i.e. 1/2 Base Height.

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