4. Rhombus QRST has diagonals intersecting at W. Point U is located on side QR and point V on diagonal RT
such that UV is perpendicular to QR.

Prove: QW•UR =WT•UV

Given:

In Rhombus QRST, diagonals QS and RT intersect at W and U∈QR and point V∈RT  such that UV⊥QR. (shown in below diagram)

To prove: QW•UR =WT•UV

Proof:

In a rhombus diagonals bisect perpendicularly,

Thus, in QRST

QW≅WS, WR ≅ WT and m∠QWR=m∠QWT=m∠RWS=m∠TWS=90°.

In triangles QWR and UVR,

             (Right angles)

             (Common angles)

By AA similarity postulate,

The corresponding sides in similar triangles are in same proportion,

                (∵ WR ≅ WT )

Hence, proved.

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