Question

PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM . MR.

NCERT Class X
Mathematics - Mathematics

Chapter _Triangles

Answer 1

WATCH DIAGRAM ARE IN ATTACHMENT
In ΔQMP and ΔQPR
      <QMP = <QPR                       (EACH = 90 degree)
           <Q = <Q                                 (COMMON)
 ⇒     ΔQMP SIMILAR ΔQPR   ....(1)  (AA similarity)
      Again in ΔPMR and ΔQPR,
          <PMR = <QPR                     (EACH = 90 degree)
           <R = < R                              (COMMON)
 ⇒    ΔRMP similar ΔQPR     .......(2) (AA similarity)
From (1) and (2) we get
                    ΔQMP similar ΔPMR
Therefore, 
              The corresponding sides are proportional 
                                     QM/PM = PM/RM
 ⇒              QM.RM = PM. PM          (BY CROSS-MULTIPLICATION) 
 ⇒               PM² = QM.RM
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Answer 2

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