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Question

PQR is a triangle right angled at P and M is a point on QR such that PMQR.

Show that PM2=QM.MR


A
PM2=QM.MR
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B
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C
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D
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Solution

Given:

In PQR,P=90

PMQR

To prove: PM2=QM.MR

Proof:

In PQR,

Let Q=θ

R=90θ [Since, angle sum property of triangle]

Now, it is given that, PMQR

In PMR,

RMP=90

RPM=90θ [Since, angle sum property of triangle]

Similarly in PMQ

PQM=θ [Assumed]

QMP=90

QPM=90θ

Now, In PQM and RPM

PQM=RPM=θ

PMQ=PMR=90

By A.A Similarity criterion,

PQMRPM

Corresponding sides are in the same ratio.

QMMP=PMMR

QMPM=PMMR

PM2=QM.MR

Hence, proved.


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