Two chords AB and AC of a circle subtends angles equal to 90º and 150º, respectively at the centre. Find ∠BAC, if AB and AC lie on the opposite sides of the centre.
Solution:
Given, two chords AB and AC of a circle subtends angles equal to 90º and 150º at the centre.
AB and AC lie on the opposite sides of the centre.
We have to find ∠BAC
Considering triangle BOA,
OA = OB = radius of the circle
We know that the angles opposite to the equal sides are equal.
So, ∠OAB = ∠OBA ----------------------- (1)
In triangle OAB,
By angle sum property of a triangle,
∠OBA + ∠OAB + ∠AOB = 180°
From (1),
∠OAB + ∠OAB + ∠AOB = 180°
Given, ∠AOB = 90°
∠OAB + ∠OAB + 90° = 180°
2∠OAB = 180° - 90°
2∠OAB = 90°
∠OAB = 45°
In triangle AOC,
AO = OC = radius of the circle
We know that the angles opposite to the equal sides are equal.
So, ∠OCA = ∠OAC --------------------------- (2)
By angle sum property of a triangle,
∠AOC + ∠OAC + ∠OCA = 180°
Given, ∠AOC = 150°
150° + ∠OAC + ∠OCA = 180°
From (2),
∠OAC + ∠OAC = 180° - 150°
2∠OAC = 30°
∠OAC = 15°
Now, ∠BAC = ∠OAB + ∠OAC
= 45° + 15°
= 60°
Therefore, ∠BAC = 60°
✦ Try This: ABCD is a cyclic quadrilateral, in which BC is parallel to AD, ∠ADC = 110° and ∠BAC = 50°. What is the value of ∠DAC?
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10
NCERT Exemplar Class 9 Maths Exercise 10.3 Problem 9
Two chords AB and AC of a circle subtends angles equal to 90º and 150º, respectively at the centre. Find ∠BAC, if AB and AC lie on the opposite sides of the centre
Summary:
Two chords AB and AC of a circle subtends angles equal to 90º and 150º, respectively at the centre. If AB and AC lie on the opposite sides of the centre, then ∠BAC = 60°
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