Question

ABCD is a quadrilateral. If P, Q, R, S are the points of trisection of the sides AB, BC, CD and DA respectively and are adjacent to A and C, then PQRS is a

• A. square
• B. rectangle
• C. rhombus
• D. parallelogram

Answer 1

The correct option is D parallelogram

Join $AC$

In $△ABC,$ we have

$\Rightarrow$ $\frac{AP}{PB}=\frac{1}{2}$

$\Rightarrow$ And $\frac{CQ}{QB}=\frac{1}{2}$

$\Rightarrow$ $\frac{AP}{AB}=\frac{CQ}{QB}$

$\therefore$ By using converse of basic proportionality theorem, we have

$PQ\parallel AC$ ------ ( 1 )

In $△ADC,$ we have

$\Rightarrow$ $\frac{DS}{AS}=\frac{2}{1}$

And $\frac{DR}{RC}=\frac{2}{1}$

$\Rightarrow$ $\frac{DS}{AS}=\frac{DR}{RC}$

$\therefore$ By using converse of basic proportionality theorem, we have

$SR\parallel AC$ ------ ( 2 )

Comparing ( 1 ) and ( 2 ), we get

$\Rightarrow$ $SR\parallel PQ$

Similarly, we can prove

$\Rightarrow$ $SP\parallel RQ$

$\therefore$ $PQRS$ is a parallelogram.