Question

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.

Easy
Updated on : 2022-09-05
Solution
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Here, we are joining A and C.
In ABC
P is the mid point of AB
Q is the mid point of BC
PQAC [Line segments joining the mid points of two sides of a triangle is parallel to AC(third side) and also is half of it]
PQAC
In ADC
R is mid point of CD
S is mid point of AD
RSAC [Line segments joining the mid points of two sides of a triangle is parallel to third side and also is half of it]
RSAC
So, PQRS and PQRS [one pair of opposite side is parallel and equal]
In APS & BPQ
APBP [P is the mid point of AB)
PASPBQ(All the angles of rectangle are )
ASBQ
APSBPQ(SAS congruency)
PSPQ
BSPQ & PQRS (opposite sides of parallelogram is equal)
PQRSPSRQ[All sides are equal]
PQRS is a parallelogram with all sides equal
So PQRS is a rhombus.

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