Question

In a square diagonals are equal prove that its a rhombus​

Answer 1

Answer:

Each diagonal cuts the other into two equal parts and the angle where they cross is always a right angle. All the sides are also equal, i.e. AB = BC = CD = DA. Quadrilateral ABCD is a square. Therefore, we proved that a rhombus with equal diagonals is a square.

Answer 2

Answer:

. In rhombus, the diagonals bisect each other at right angles. Prove that the vertices are also right angles.

From the figure, let us say that ABCD is a rhombus.

Let AC and BD be the diagonals of the rhombus. It is said that the rhombus has equal diagonals, i.e. AC = BD.

We know that diagonals of rhombus bisect each other. Thus we can say that AC = BD.

If AC = BD, then

AO = BO = CO = DO…..(1)

Consider the point where the diagonals intersect at O.

Let us consider the ΔAOB

.

From equation (1) we can say that AO = BO and ∠AOB=90∘

.

In rhombus, the diagonals bisect each other at right angles. Each diagonal cuts the other into two equal parts and the angle where they cross is always a right angle.

∴∠AOB=90∘

.

Thus we can say that ∠OAB=∠OBA=90∘2=45∘

Similarly in ΔAOD

, ∠OAD=∠ODA=90∘2=45∘

Thus from the figure,

∠A=∠OAB+∠OAD=45∘+45∘=90∘

Similarly, from the figure,

∠B=∠C=∠D=90∘

∴

All the angles, ∠A=∠B=∠C=∠D=90∘

All the sides are also equal, i.e. AB = BC = CD = DA.

∴

Quadrilateral ABCD is a square.

Therefore, we proved that a rhombus with equal diagonals is a square.

Note:In case of a square, we know the general properties, that all sides of a square are equal in length. Thus the diagonals of a square are equal in length and diagonals bisect each other. The angles are 90∘

. Thus the square has 4 congruent sides and 4 right angles.