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  2. PQR is a triangle right angled at P and M is a point on QR such that PM perpendicular to QR. Show that PM square equals QM . MR.

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PQR is a triangle right angled at P and M is a point on QR such that PM perpendicular to QR. Show that PM square equals QM . MR.

Q2   PQR is a triangle right angled at P and M is a point on QR such that PM \perp QR. Show that
        PM ^2 = QM . MR .

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S seema garhwal

Let \angle MPR   be x

In \triangle MPR,

\angle MRP=180 \degree-90 \degree-x

\angle MRP=90 \degree-x

Similarly,

In \triangle MPQ,

\angle MPQ=90 \degree-\angle MPR

\angle MPQ=90 \degree-x

\angle MQP=180 \degree-90 \degree-(90 \degree-x)=x

In \triangle QMP\, and\, \triangle PMR,

\angle MPQ\, =\angle MRP

\angle PMQ\, =\angle RMP

\angle MQP\, =\angle MPR

\triangle QMP\, \sim \triangle PMR,                (By AAA)

\frac{QM}{PM}=\frac{MP}{MR}

\Rightarrow PM^2=MQ\times MR

Hence proved 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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