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प्रश्न
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
पर्याय
a rhombus
a rectangle
a square
any parallelogram
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उत्तर
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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संबंधित प्रश्न
ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal. Show that:
- SR || AC and SR = `1/2AC`
- PQ = SR
- PQRS is a parallelogram.
ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
Fill in the blank to make the following statement correct:
The triangle formed by joining the mid-points of the sides of a right triangle is
In the given figure, seg PD is a median of ΔPQR. Point T is the mid point of seg PD. Produced QT intersects PR at M. Show that `"PM"/"PR" = 1/3`.
[Hint: DN || QM]
Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.
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Prove that : (i) AP = 2AR
(ii) BC = 4QR
In Δ ABC, AD is the median and DE is parallel to BA, where E is a point in AC. Prove that BE is also a median.
In the given figure, ABCD is a trapezium. P and Q are the midpoints of non-parallel side AD and BC respectively. Find: DC, if AB = 20 cm and PQ = 14 cm
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In ΔABC, D and E are the midpoints of the sides AB and AC respectively. F is any point on the side BC. If DE intersects AF at P show that DP = PE.
