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
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