Question

ABCD is a quadrilateral in which $P,Q,R$ and $S$ are mid -point of the side $AB,BC,CD$ and $DA$ . $AC$ is a diagonal. then

• A. $SR\parallel AC$ and $SR=\frac{1}{2}AC$
• B. $PQ\ne SR$
• C. $RS=AB$
• D. None of these

Answer 1

Answer: (A)

$SR\parallel AC$ and $SR=\frac{1}{2}AC$

In $\Delta ADC$ and $\Delta SDR$

$\angle D$ is common

$\frac{AD}{SD}=\frac{DC}{DR}=2$ {S and R are midpoints of AD and DC]

Thus $\Delta ADC\sim \Delta SDR$ by SAS Criteria

That is corresponding angles are equal and the ratio of sides is the same

$\therefore \angle DSR=\angle DAC$ and $\angle SRD=\angle ACD$

i.e. Corresponding angles in line $SR$ and $AC$.

Since corresponding angles are equal $\therefore SR\parallel AC$

And also the ratio of sides is the same

$\therefore \frac{AD}{SD}=\frac{DC}{DR}=\frac{AC}{SR}=2$

i.e.$\frac{AC}{SR}=2\Rightarrow SR=\frac{1}{2}AC$