Question

X and Y are the mid-points of opposite sides AB and DC of a parallelogram ABCD. AY and DX are joined intersecting at P, CX and BY are joined intersecting at Q. Show that quad. PXQY is a parallelogram.​

Answer 1

Answer:

ABCD is the given parallelogram.

AB=CD ..... (Opposite sides of parallelogram are equal)

So,

1/2AB= 1/2CD

AX=CY ...... (as X is the mid point of AB and Y is the mid point of CD)

AX∣∣CY ......... (We know AB||CD, as ABCD is the parallelogram)

Quadrilateral AXCY is a prallelogram, as a pair of opposite sides is equal and parallel.

So, XC=YA and XC∥YA

XQ∥YP ...... (i) (As XQ is part of line XC and YP is part of line YA)

We have ABCD as the given parallelogram

Now, AB=CD ....... (Opposite sides of parallelogram are equal)

So, 1/2AB= 1/2CD

DY=XB ....... (As X is the mid point of AB and Y is the mid point of CD)

DY∥XB ..... (We know AB∥CD, as ABCD is the parallelogram)

Quadrilateral DXBY is a prallelogram, as a pair of opposite sides is equal and parallel.

So, DX=YB and DX∣∣YB

PX∣∣YQ ...... (ii) (As XP is part of line XD and YQ is part of line YB)

From (i) and (ii) we get,

PXQY is also a parallelogram