Question

ABCD is a parallelogram in which x and y are midpoints of a b and CD respectively a y and X are join which intersect each other at P B Y and Z X has also join which intersect each other at Q show that PHQ why is a parallelogram

Answer 1

We form our diagram from given information , As :


We have ABCD as the given parallelogram.

Now, AB = CD [Opposite sides of parallelogram are equal]

so, 1/2 AB = 1/2 CD

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

AX | | CY ( We know AB | | CD , as ABCD as the given parallelogram)

Quadrilateral AXCY is a parallelogram [In a quad, if a pair of opposite sides is equal and ||, then it is a ||gm]

So, XC = YA and XC | | YA

XQ | | YP ------------- ( 1 ) ( 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, 1/2 AB = 1/2 CD

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

DY | | XB ( We know AB | | CD , as ABCD as the given parallelogram)

Quadrilateral DXBY is a parallelogram [In a quad, if a pair of opposite sides is equal and ||, then it is a ||gm]

So, DX = YB and DX | | YB

PX | | YQ ------------- ( 2) ( As PX is part of line DX and YQ is part of line YB )

From equation (1) and (2) , we get

PXQY is also a parallelogram . ( Hence proved )