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# Show that the quadrilateral formed by joining the mid-points the sides of a rhombus, taken in order, form a rectangle.

**Solution:**

Consider a __rhombus__ ABCD

The points P, Q, R and S are the midpoints of the sides AB, BC, CD and AD.

We have to show that PQRS is a __rectangle__.

Join the __diagonals__ AC and BD of the rhombus ABCD.

Considering the triangle ABD,

Since S and P are the midpoints of the sides AD and AB.

SP || BD ----------------- (1)

SP = 1/2 BD --------------- (2)

Similarly, RQ || BD

RQ = 1/2 BD -------------- (3)

From (2) and (3),

SP = RQ

Also, SP || RQ

Therefore, PQRS is a __parallelogram__

We know that the diagonals of a rhombus are perpendicular.

So, AC⊥ BD -------------- (4)

Considering triangle BAC,

PQ || AC --------------- (5)

From (1), (4) and (5),

SP ⊥ PQ

i.e.,∠SPQ = 90°

We know that a rectangle is a __quadrilateral__ with four right angles. The opposite sides are __parallel__ and equal to each other.

Therefore, PQRS is a rectangle.

**✦ Try This: **Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a rectangle is a rhombus.

**☛ Also Check:** NCERT Solutions for Class 9 Maths Chapter 8

**NCERT Exemplar Class 9 Maths Exercise 8.4 Sample Problem 3**

## Show that the quadrilateral formed by joining the mid-points the sides of a rhombus, taken in order, form a rectangle.

**Summary:**

It is shown that the quadrilateral formed by joining the mid-points the sides of a rhombus, taken in order, forms a rectangle as the angle is equal to 90 degrees

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