Question

ABCD is a parallelogram. Two lines l and m are parallel to AD. Line I meets AB and CD at P and S respectively. Line m meets AB and CD at Q and R respectively. X is any point on CD between R and S. If ar($\Delta$DPX)$+$ar($\Delta$CQX)$=$k ar(ABCD), find k.

• A. $\frac{2}{3}$
• B. $\frac{3}{2}$
• C. $\frac{1}{2}$
• D. $\frac{1}{3}$

Answer 1

Answer: (C)

$\frac{1}{2}$

In parallelogram $ABCD$, Opposite sides are congruent

$AD=BC$

and $\because AD||PS||QR||BC$

$\therefore PS=AD=QR=BC$

$ar\left(△DPX\right)+ar\left(△CQX\right)=\frac{1}{2}PS\times DX+\frac{1}{2}QR\times CX=\frac{1}{2}\times AD(DX+XC)=\frac{1}{2}AD\times CD=\frac{1}{2}ar\left(ABCD\right)$

$\therefore k=\frac{1}{2}$