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aliina [53]
3 years ago
13

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

Mathematics
1 answer:
ad-work [718]3 years ago
8 0

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.

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