Sides \(AB\) and \(AC\) and median \(AD\) of a \(\Delta ABC\) are respectively proportional to sides \(PQ\) and \(PR\) and median \(PM\) of another \(\Delta PQR\). Show that \(\Delta ABC \sim \Delta PQR\).
Proof:
Given, that in triangles \(ABC\) and \(PQR\) in which \(AD\) and \(PM\) are medians such that \(\frac{AB}{PQ} = \frac{AC}{PR} = \frac{AD}{PM}\).
Construction : Produce \(AD\) to \(E\) so that \(AD = DE\). Join \(CE\).
Similarly, produce \(PM\) to \(X\) such that \(PM = MX\). Also, join \(RX\).
In \(\Delta ABD\) and \(\Delta CDE\),
(by construction)
(\(AD\) is the median)
(vertically opposite angles)
Thus, \(\Delta ABD \cong \Delta CED\) (by \(SAS\) congruence criterion)
(by CPCT) - - - - (i)
In \(\Delta PQM\) and \(\Delta MNR\),
(by construction)
(\(PM\) is the median)
(vertically opposite angles)
\(\Delta PQM \cong \Delta MNR\) (by \(SAS\) congruence criterion)
(by CPCT) - - - - (ii)
Now,
(from (i) and (ii))
Thus, \(\Delta ACE \sim \Delta PRN\) (by \(SSS\) similarity criterion)
Therefore, .
Similarly, .
- - - - (iii)
In \(\Delta ABC\) and \(\Delta PQR\), we have
\(\frac{AB}{PQ} = \frac{AC}{PR}\) (Given)
(from (iii))
Therefore, \(\Delta ABC \sim \Delta PQR\) (by \(SAS\) similarity criterion).
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