Question

what shape is expected by the join of PQRS if ABCD forms a rhombus?​

Answer 1

Answer:

figure,

ABCD is a rhombus and P, Q, R, S are mid-point of AB, BC, CD and DA.

In △ABD,

Since, P and S are mid-point of AB and AD

∴ By Mid-point theorem,

P ∣∣BD; PS =

2

1

BD ----- (1)

In △BCD,

Since, R and Q are mid-point of sides CD & CB

∴ By Mid-point theorem,

RQ∣∣BD; RQ =

2

1

BD ------(2)

From (1) & (2),

PS∣∣QR and PS = QR -------(3)

Similarly,

PQ∣∣RS and PQ = RS -------(4)

From (3) and (4) we get,

Quadrilateral PQRS is a parallelogram.

The diagonals of a rhombus bisect each other at right angles.

In quadrilateral OMQL,

since, QR∣∣PS

∴ OM∣∣QL; OM = QL

As PQ∣∣RS

∴ QM∣∣OL; QM = OL

∴ Quadrilateral OMQL is a parallelogram.

As, ∠O = 90

∘

∠MQL = ∠MOL = 90

∘

---(opposite angles of parallelogram are equal)

∴ PQRS is a rectangle.

solution