Question

Pqrs is a rectangle whose length is 2 times of a breath and yellow the midpoint of ptm with p and q are centres of drought true

Answer 1

Answer

Given,

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)

(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)