Question

In the given figure, ABCD is a trapezium in which AB || DC and DC = 40 cm and AB = 60 cm. If X and Y are, respectively, the mid-points of AD and BC, prove that:

(i) XY = 50 cm

(ii) DCYX is a trapezium

(iii) ar (trap. DCYX) = $\frac{9}{11}$ ar (trap. (XYBA)

Answer 1

Given: ABCD IS A trapezium in which

(a) AB||DC

(b) DC = 40 cm

(c) AB = 60 cm

(d) X is the midpoint of AD

(e) Y is the midpoint of BC

To prove:

(i) XY = 50 cm

(ii) DCYX is a trapezium

(iii)

Construction: Join DY and produce it to meet AB produced at P.

Proof:

(i) In ΔBYP and ΔCYD

Y is the midpoint of BC also X is the midpoint of AD

Therefore XY||AP and

(ii) We have proved above that XY||AP

XY|| AP and AB||DC (Given in question)

XY|| DC

(iii) Since X and Y are the midpoints of AD and BC respectively.

Therefore DCYX and ABYX are of the same height say h cm.