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[tex] \underline{ \underline{ \text{question}}} : [/tex] 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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xKelvin

Answer:

See Below.

Step-by-step 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:

[tex]SK=QK\text{ and } PK = RK[/tex]

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

[tex]SK=SX+XK\text{ and } QK=QY+YK[/tex]

Since SK = QK:

[tex]SX+XK=QY+YK[/tex]

And since we are given that SX = QY:

[tex]XK=YK[/tex]

So we now have:

[tex]XK=YK\text{ and } PK=RK[/tex]

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

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