BEST is a rhombus and BE is produced to P and Q such that BP = BE = EQ, prove that
PR and QR are perpendicular to each other.
Answers
Answer:
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Step-by-step explanation:
ABCD is a rhombus.
P,Q and R are the mid-points of AB,BC and CD.
Join AC and BD.
In △DBC,
R and Q are the mid-points of DC and CB
∴ RQ∥DB [ By mid-point theorem ]
∴ MQ∥ON ---- ( 1 ) [ Parts of RQ and DB ]
Now, in △ACB,
P and Q are the mid-points of AB and BC
∴ AC∥PQ [ By midpoint theorem ]
∴ OM∥NQ ---- ( 2 ) [ OM and NQ are the parts of AC and PQ ]
From equation ( 1 ) and ( 2 )
⇒ MQ∥ON
⇒ ON∥NQ
Since, each pair of opposite side is parallel.
∴ ONQM is a parallelogram.
In ONQM,
⇒ ∠MON=90o [ Diagonals of rhombus bisect each other ]
⇒ ∠MON=∠PQR [ Opposite angles of parallelogram are equal ]
∴ ∠PQR=90o.
Hence, PQ⊥QR ---- Hence proved