ABCD is a rectangle and P,Q,R,S are the midpoints of its sides AB,BC,CD and DA respectively. Show that the quadrilateral PQRS is a

2 thoughts on “ABCD is a rectangle and P,Q,R,S are the midpoints of its sides AB,BC,CD and DA respectively. Show that the quadrilateral PQRS is a”

1. Step-by-step explanation:

   P, Q, R, S are the midpoints of AB, BC, CD and DA respectively.

   ABCD is a rectangle

   To Prove

   PQRS is a rhombus.

   Construction

   Join AC. Join the Midpoints.

   Proof

   In ΔDAC

   S and R are the midpoints of DA and DC

   ∴ By Midpoint Theorem,

   SR ║ AC →1

   SR = \frac{1}{2}

   2

   1

   AC →2

   In Δ BAC

   P and Q are the midpoints of AB and BC

   ∴ By Midpoint Theorem,

   PQ ║ AC →3

   PQ = \frac{1}{2}

   2

   1

   AC → 4

   Now,

   From 1 and 3

   SR ║PQ → 5

   From 2 and 4

   SR = PQ → 6

   From 5 and 6

   PQRS is a parallelogram (one pair of opposite sides are equal and parallel)

   Now,

   In ΔSAP and ΔQBP

   AD = BC (Opp sides of a rectangle are equal)

   \frac{AD}{2} = \frac{BC}{2}

   2

   AD

   =

   2

   BC

   (halves of equals are equal)

   AS = BQ (S and Q are midpoints)

   ∠A = ∠B = 90° (Angles of a rectangle)

   AP = BP (P is the midpoint of AB)

   ∴ ΔSAP ≅ ΔQBP by SAS congruency

   ⇒ PS = PQ (CPCT)

   ∴ PQRS Is a rhombus.

   (In a parallelogram if adjacent sides are equal, it is a rhombus, the adjacent sides her are PS and PQ)

   ———————–

   Hope It Helps!

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2. Here, we are joining A and C.

   In ΔABC

   P is the mid point of AB

   Q is the mid point of BC

   PQ∣∣AC [Line segments joining the mid points of two sides of a triangle is parallel to AC(third side) and also is half of it]

   PQ=

   2

   1

   AC

   In ΔADC

   R is mid point of CD

   S is mid point of AD

   RS∣∣AC [Line segments joining the mid points of two sides of a triangle is parallel to third side and also is half of it]

   RS=

   2

   1

   AC

   So, PQ∣∣RS and PQ=RS [one pair of opposite side is parallel and equal]

   In ΔAPS & ΔBPQ

   AP=BP [P is the mid point of AB)

   ∠PAS=∠PBQ(All the angles of rectangle are 90

   o

   )

   AS=BQ

   ∴ΔAPS≅ΔBPQ(SAS congruency)

   ∴PS=PQ

   BS=PQ & PQ=RS (opposite sides of parallelogram is equal)

   ∴ PQ=RS=PS=RQ[All sides are equal]

   ∴ PQRS is a parallelogram with all sides equal

   ∴ So PQRS is a rhombus.

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