Show that the quadrilateral formed by joining the midpoints of the consecutive sides of a square is also a square.

To Prove: Quadrilateral formed by joining the midpoints of the consecutive sides of a square is also a square.

Given: Midpoints of a square are joined.

Concept Used:

SAS Theorem: If two sides and one angle of a triangle are equal to two sides and one angle of another triangle, triangles are congruent.

Diagram:

Proof:

From the figure, we need to prove that EFGH is a square.

In ΔAEF and ΔBFG,

∠EAF = FBG [All angles of a square are equal]

AD = AB [Side of the square]

AE = AF = BF = BG

Thus,

ΔAEF and ΔBFG are congruent.

So,

EF = FG

And similarly, we can see that,

EF = FG = GH = HE

And,

∠AEF = DEH = 45˚

So,

∠AEF + ∠DEH + ∠HEF = 180˚ [Sum of angles on a straight line = 180˚]

Therefore,

∠DEH = 180˚ - 90˚

∠DEH = 90˚

Similarly, all the angles of quadrilateral EFGH is a right angle.

As all the sides of quadrilateral EFGH are equal and all the angles are right angles. EFGH is a square.

Hence, Proved.