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

Answer:

Given:

In the rhombus QRST, diagonals QS and RT cross at points W and U ∈ QR, while point V ∈ RT is such that UV⊥QR. (as illustrated in the diagram below)

To prove: QW•UR = WT•UV

Proof:

In a rhombus, the diagonals meet at right angles and bisect one another,

Thus, in QRST

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

Considering triangles QWR and UVR,

             (Right angles)

             (Shared angles)

By the AA similarity postulate,

The sides corresponding in similar triangles maintain proportionality,

                (∵ WR ≅ WT )

Thus, it is proved.