Question

in the adjoining figure ABCD is a equatorial in which a d equals to BC and p q r s r the midpoints of ab BC CD and ac respectively prove that pqrs is a rhombus​

Answer 1

Step-by-step explanation:

In triangle BAD, P and S are midpoints of AB and BD

= PS || AD and PS=1/2AD ---(1) [ By BPT]

Similarly in Triangle CAD,

= OR || AD and QR= 1/2AD ---(2)

From (1) and (2), we get

= PS || QR and PS=QR= 1/2AD ---(3)

In Triangle BDC, we get

= SR || BC and SR= 1/2BC ---(4)

And in Triangle ABC

PQ || BC and PQ= 1/2BC ---(5)

= PQ || BC and PQ= 1/2BC ---(6) [From (4) and (5)]

Therefore Square PQRS is a parallelogram.

Now, AD = BC

Therefore 1/2AD = 1/2BC

Therefore PS = QR = PQ = SR [From (3) and (6)]

Therefore Square PQRS is a rhombus.

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