Question

Heya!


`\underline{ \underline{ \text{question}}} :` In the adjoining figure , PQRS is a parallelogram and X , Y are points on the diagonal QS such that SX = QY. Prove that the quadrilateral PXRY is a parallelogram.


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Answer 1

See Below.

Explanation:

We are given that PQRS is a parallelogram, where X and Y are points on the diagonal QS such that SX = QY.

And we want to prove that quadrilateral PXRY is a parallelogram.

Since PQRS is a parallelogram, its diagonals bisect each other. Let the center point be K. In other words:

`SK=QK\text{ and } PK = RK`

SK is the sum of SX and XK. Likewise, QK is the sum of QY and YK:

`SK=SX+XK\text{ and } QK=QY+YK`

Since SK = QK:

`SX+XK=QY+YK`

And since we are given that SX = QY:

`XK=YK`

So we now have:

`XK=YK\text{ and } PK=RK`

Since XY bisects RP and RP bisects XY, PXRY is a parallelogram.