Question

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
​

Answer 1

Answer:

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,

$m\angle QWR=m\angle VUR$              (Right angles)

$m\angle WRQ=m\angle VRU$              (Common angles)

By AA similarity postulate,

$\triangle QWR\sim \triangle VUR$

The corresponding sides in similar triangles are in same proportion,

$\implies \frac{QW}{VU}=\frac{WR}{UR}$

$QW\times UR=WR\times VU$

$QW\times UR=WT\times UV$                 (∵ WR ≅ WT )

Hence, proved.