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# Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square.

**Solution:**

Consider a __square__ ABCD

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

We have to show that PQRS is a square.

Join the diagonals AC and BD of the square ABCD

We know that all the sides of the square are equal in length

So, AB = BC = CD = AD

The __midpoint theorem__ states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

Considering triangle ADC,

S and R are the midpoints of AD and DC

By midpoint theorem,

SR || AC

SR = 1/2 AC --------------- (1)

Considering triangle ABC,

P and Q are the midpoints of AB and BC

By midpoint theorem,

PQ || AC

PQ = 1/2 AC ----------------- (2)

Comparing (1) and (2),

SR = PQ = 1/2 AC ------------ (3)

Considering triangle BAD,

SP || BD

By midpoint theorem,

SP = 1/2 BD ----------------- (5)

Comparing (4) and (5),

SP = RQ = 1/2 BD ----------- (6)

We know that the diagonals of a square bisect each other at right angle.

So, AC = BD

Dividing by 2 on both sides,

1/2 AC = 1/2 BD

From (3) and (6),

SR = PQ = SP = RQ

Considering __quadrilateral__ OERF,

OE || FR

OF || ER

∠FOE = ∠FRE = 90°

Therefore, PQRS is a square.

**✦ Try This: **Can a quadrilateral ABCD be a parallelogram if ∠D + ∠B = 180°?

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

**NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 11**

## Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square.

**Summary:**

Square is a regular quadrilateral, which has all the four sides of equal length and all four angles are also equal. The angles of the square are at right-angle or equal to 90-degrees. It is shown that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square

**☛ Related Questions:**

- E and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD. Pro . . . .
- Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle
- P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through t . . . .

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