Mathematics
P, Q and R are mid-points of sides AB, BC and CD respectively of a rhombus ABCD. Show that PQ is perpendicular to QR.
Mid-point Theorem
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Answer
Join diagonals of rhombus AC and BD.
We know that,
Diagonals of rhombus intersect at 90°.
By mid-point theorem,
The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it.
In △ ABC,
P and Q are mid-points of sides AB and BC respectively.
∴ PQ || AC (By mid-point theorem)
In △ BCD,
R and Q are mid-points of sides CD and BC respectively.
∴ QR || BD (By mid-point theorem)
Since, AC ⊥ BD and PQ || AC and QR || BD.
∴ PQ ⊥ QR.
Hence, PQ is perpendicular to QR.
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