# AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters, (ii) ABCD is a rectangle

**Solution:**

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle and the sum of either pair of opposite angles of a cyclic quadrilateral is 180º.

We prove the statement using Side-Angle-Side (SAS criteria) and Corresponding parts of congruent triangles (CPCT).

Let AC and BD be two chords intersecting at O.

In ΔAOB and ΔCOD,

OA = OC (Given)

OB = OD (Given)

∠AOB = ∠COD (Vertically opposite angles)

Hence, ΔAOB ≅ ΔCOD (SAS congruence rule)

AB = CD (By CPCT)

Similarly, it can be proved that ΔAOD ≅ ΔCOB

Hence, AD = CB (By CPCT)

Since in quadrilateral ABCD, opposite sides are equal in length, ABCD is a parallelogram.

We know that opposite angles of a parallelogram are equal.

Therefore, ∠A = ∠C

However, ∠A + ∠C = 180° (ABCD is a cyclic quadrilateral)

∠A + ∠A = 180°

2∠A = 180°

∴ ∠A = 90°

ABCD is a parallelogram and one of its interior angles is 90°, therefore, it is a rectangle.

∠A is the angle subtended by chord BD, ∠A = 90°, therefore, BD should be the diameter of the circle

Similarly, AC is the diameter of the circle.

Thus, (i) AC and BD are diameters, and (ii) ABCD is a rectangle, proved

**Video Solution:**

## AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters, (ii) ABCD is a rectangle

### Maths NCERT Solutions Class 9 - Chapter 10 Exercise 10.6 Question 7:

**Summary:**

AC and BD are chords of a circle which bisect each other. We have proved that AC and BD are diameters and also that ABCD is a rectangle