Question

The middle points of the parallel sides AB and CD of a parallelogram ABCD are P and Q respectively. If AQ and CP divide the diagonal BD into three parts BX, XY and YD, then which one of the following is correct? 

(a) BX ≠ XY ≠ YD 

(b) BX = YD ≠ XY 

(c) BX = XY = YD 

(d) XY = 2BX

Answer 1

(c) BX = XY = DY.

ABCD is a parallelogram. 

⇒ AB = CD and AB || CD 

⇒ AP = QC and AP || QC 

⇒ APCQ is a parallelogram. 

⇒ AQ || PC

By intercept theorem in ΔDXC,

we have \(\frac{DY}{YX}=\frac{DQ}{QC}=1\)

⇒ DY = YX 

Also using intercept theorem in ΔABY

\(\frac{BX}{XY}\) = \(\frac{BP}{PA}\) = \(\frac{1}{1}\)⇒ BX = XY.

∴ BX = XY = DY.