Question

ABCD is a rhombus with P, Q and R as mid-points of AB, BC and CD respectively.
Prove that PQ is perpendicular to QR.

Please solve... Thanks in advance !​

Answer 1

Answer:

Angle PQR = 90 degree

•°• PQ is perpendicular to QR

Step-by-step explanation:

GIVEN :

ABCD is a rhombus with P, Q and R as mid points of AB, BC and CD and centre O.

To prove : PQ is perpendicular to QR,

we must prove angle PQR = 90

Construction : Join AC and BD. Join PQ with intersecting point M and QR with N.

PROOF : In triangle DBC

RQ // DB (since R and Q are the mid points of DC and CB)

MQ // ON (parts of RQ and DB )

Now in Triangle ABC,

P and Q are the mid points of AB and BC

•°•AC // PQ

From eq. (1 ) & (2)

Each pair of opp. sides are parallel.

In QMON,

angle MON = 90 degree (diagonals of rhombus bisect each other at 90 degree )

•°• angle MON = angle PQR (opp. angles of parallelogram are equal )

°•° angle PQR = 90 degree.

•°• PQ is perpendicular to QR.

Hence, proved.

Answer 2

Answer:

Angle PQR = 90 degree

•°• PQ is perpendicular to QR

Step-by-step explanation:

GIVEN :

ABCD is a rhombus with P, Q and R as mid points of AB, BC and CD and centre O.

To prove : PQ is perpendicular to QR,

we must prove angle PQR = 90

Construction : Join AC and BD. Join PQ with intersecting point M and QR with N.

PROOF : In triangle DBC

RQ // DB (since R and Q are the mid points of DC and CB)

MQ // ON (parts of RQ and DB )

Now in Triangle ABC,

P and Q are the mid points of AB and BC

•°•AC // PQ

From eq. (1 ) & (2)

Each pair of opp. sides are parallel.

In QMON,

angle MON = 90 degree (diagonals of rhombus bisect each other at 90 degree )

•°• angle MON = angle PQR (opp. angles of parallelogram are equal )

°•° angle PQR = 90 degree.

•°• PQ is perpendicular to QR.

Hence, proved.