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Question
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
Options
a rhombus
a rectangle
a square
any parallelogram
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Solution
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is a rectangle.
Explanation:
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
Join AC, RP and SQ
In ∆ABC,
P is midpoint of AB and Q is midpoint of BC
∴ By midpoint theorem,
PQ || AC and PQ = `1/2` AC ...(1)
Similarly,
In ∆DAC,
S is midpoint of AD and R is midpoint of CD
∴ By midpoint theorem,
SR || AC and SR = `1/2` AC ...(2)
From (1) and (2),
PQ || SR and PQ = SR
⇒ PQRS is a parallelogram
ABQS is a parallelogram
⇒ AB = SQ
PBCR is a parallelogram
⇒ BC = PR
⇒ AB = PR ...[∵ BC = AB, sides of rhombus]
⇒ SQ = PR
∴ Diagonals of the parallelogram are equal.
Hence, it is a rectangle.
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