In Fig. 10.19, AB and CD are two chords of a circle intersecting each other at point E. Prove that ∠AEC = 1/2 (Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre).
Solution:
Given, AB and CD are two chords of a circle intersecting each other at point E.
We have to prove that ∠AEC = 1/2 (Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre).
Extend the line OD and OB at the points l and H on the circle.
Join AC
We know that the angle subtended by an arc at the centre of a circle is twice the angle subtended by it at the remaining part of the circle.
∠1 = 2∠6 ------------------- (1)
∠3 = 2∠7 ------------------ (2)
In triangle AOC,
OC = OA = radius of the circle
We know that the angles opposite to the equal sides are equal.
∠OCA = ∠4
By angle sum property of a triangle,
∠AOC + ∠OCA + ∠4 = 180°
∠AOC + ∠4 + ∠4 = 180°
∠AOC + 2∠4 = 180°
∠AOC = 180° - 2∠4 --------------------- (3)
In triangle OCD,
OC = OD
We know that the angles opposite to the equal sides are equal.
∠ECO = ∠6 ---------------------------------- (4)
Similarly, in triangle ∠5 = ∠7 ----------- (5)
In triangle AEC,
By angle sum property of a triangle,
∠AEC + ∠EAC + ∠ECA = 180°
∠AEC = 180° - (∠EAC + ∠ECA)
Now, ∠AEC = 180° - [(∠ECO + ∠OCA) + (∠CAO + ∠OAE)]
From (4),
∠AEC = 180° - [(∠6 + ∠4) + (∠4 + ∠5)]
∠AEC = 180° - (2∠4 + ∠6 + ∠5)
∠AEC = 180° - 2∠4 - ∠6 - ∠5
From (3),
∠AEC = ∠AOC - ∠6 - ∠5
From (5),
∠AEC = ∠AOC - ∠6 - ∠7
From (1) and (2),
∠AEC = ∠AOC - 1/2 ∠1 - 1/2 ∠3
By adding and subtracting ∠2/2,
∠AEC = ∠AOC - ∠1/2 - ∠2/2 - ∠3/2 + ∠2/2
= ∠AOC - 1/2 (∠1 + ∠2 + ∠3) + ∠2/2
We know that the vertically opposite angles are equal
∠2 = ∠8
So, ∠AEC = ∠AOC - 1/2 (∠1 + ∠2 + ∠3) + ∠8/2
= ∠AOC - ∠AOC/2 + ∠DOB/2
∠AEC = ∠AOC/2 + ∠DOB/2
∠AEC = 1/2 (∠AOC + ∠DOB)
Therefore, ∠AEC = 1/2 (Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre).
✦ Try This: If an arc of a circle of radius 14 cm subtends an angle of 60° at the centre, then the length of the arc is 44/3 cm.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10
NCERT Exemplar Class 9 Maths Exercise 10.4 Problem 8
In Fig. 10.19, AB and CD are two chords of a circle intersecting each other at point E. Prove that ∠AEC = 1/2 (Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre).
Summary:
In Fig. 10.19, AB and CD are two chords of a circle intersecting each other at point E. It is proven that ∠AEC = 1/2 (Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre)
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