# Show that the diagonals of a square are equal and bisect each other at right angles

**Solution:**

Given: The quadrilateral is a square.

To prove: Diagonals of a square are equal and bisect each other at right angles.

Thus, we have to prove AC = BD, OA = OC, OB = OD, and ∠AOB = 90°

Let ABCD be a square.

Let the diagonals AC and BD intersect each other at a point O.

In ΔABC and ΔDCB,

AB = DC (Sides of a square are equal to each other)

∠ABC = ∠DCB (All interior angles are of 90^{o})

BC = CB (Common side)

∴ ΔABC ≅ ΔDCB (By SAS congruence rule)

∴ AC = DB (By CPCT) ------------ (1)

Hence, the diagonals of a square are equal in length.

In ΔAOB and ΔCOD,

∠AOB = ∠COD (Vertically opposite angles)

∠ABO = ∠CDO (Alternate interior angles)

AB = CD (Sides of a square are always equal)

∴ ΔAOB ≅ ΔCOD (By AAS congruence rule)

∴ AO = CO and OB = OD (By CPCT) ------------ (2)

Hence, the diagonals of a square bisect each other.

In ΔAOB and ΔCOB,

As we had proved that diagonals bisect each other,

Therefore, AO = CO

AB = CB (Sides of a square are equal)

BO = BO (Common)

∴ ΔAOB ≅ ΔCOB (By SSS congruency)

∴ ∠AOB = ∠COB (By CPCT) --------- (3)

However, ∠AOB + ∠COB = 180° (Linear pair)

2∠AOB = 180° [From equation (3)]

∠AOB = 90° ------------- (4)

Hence, from equation (1), (2) and (4), we see that the diagonals of a square are equal and bisect each other at right angles

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

**Video Solution:**

## Show that the diagonals of a square are equal and bisect each other at right angles.

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 4

**Summary:**

Thus, we have proved that the diagonals of a square are equal and bisect each other at right angles.

**☛ Related Questions:**

- Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
- Diagonal AC of a parallelogram ABCD bisects ∠A (see the given figure). Show that i) it bisects ∠C also, ii) ABCD is a rhombus.
- ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.
- ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:(i) ABCD is a square(ii) diagonal BD bisects ∠B as well as ∠D.