Question

$S$ and $T$ are respectively the mid points of equal sides $PQ$ and $PR$ of $△PQR$. Show that $Qt=RS$.

Answer 1

In $△PQR$,

$PQ=PR$ ------ [ Given ]

$△PQR$ is an isosceles triangle.

$\therefore$ $\angle PQR=\angle PRQ$ [ Angles of isosceles triangle ]

i.e $\angle SQR=\angle TRQ$ ------ ( 1 )

Here, $S$ and $T$ are mid-point of $PQ$ and $PR$

$\therefore$ $SQ=TR$ ----- ( 2 )

Now, in $△SQR$ and $△TRQ$

$\Rightarrow$ $SQ=TR$ [ From ( 2 ) ]

$\Rightarrow$ $\angle SQR=\angle TRQ$ [ From ( 1 ) ]

$\Rightarrow$ $QR=RQ$ [ Common side ]

$\therefore$ $△SQR\cong △TRQ$ [ By $SAS$ criteria ]

$\therefore$ $RS=QT$ [ By c.p.c.t ]