ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the midpoints of AB, AC, CD and BD respectively, show that PQRS is a rhombus.
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Given: ABCD is a quadrilateral in which AD = BC. P, Q, R, S are the midpoint of AB, AC, CD and BD.
To prove: PQRS is a rhombus
Proof: In $\triangle\text{ABC},$
Since P and Q are mid point of AB and AC
Therefore,PQ || BC and $\text{PQ}=\frac{1}{2}\text{BC}=\frac{1}{2}\text{DA}$ (Mid-point theorem)
Similarly,
In $\triangle\text{CDA},$
Since R and Q are mid point of CD and AC
Therefore, RQ || DA and $\text{PQ}=\frac{1}{2}\text{DA}$
In $\triangle\text{BDA},$
Since S and P are mid point of BD and AB
Therefore, SP || DA and $\text{SP}=\frac{1}{2}\text{DA}$
In $\triangle\text{CDB},$
Since S and P are mid point of BD and CD
Therefore, SR || BC and $\text{SR}=\frac{1}{2}\text{BC}=\frac{1}{2}\text{DA}$
$\therefore\text{SP }||\text{ RQ}$ and $\therefore\text{PQ }||\text{ SR}$ and $\text{PQ}=\text{RQ}=\text{SP}=\text{SR}$
Hence, PQRS is a rhombus.
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