# 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.

Let's construct a figure according to the given question.

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 [Since, angle in a semicircle is a right angle]

Similarly, AC is the diameter of the circle.

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

**☛ Check: **NCERT Solutions for Class 9 Maths Chapter 10

**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

**☛ Related Questions:**

- Prove that the line of centers of two intersecting circles subtends equal angles at the two points of intersection.
- Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its center. If the distance between AB and CD is 6 cm, find the radius of the circle.
- The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the center, what is the distance of the other chord from the center ?
- Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the center.

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