MathematicsIn the given figure, two chords AB and AC of a circle subtends angles equal to 90oand 150o, respectively at the centre. Which of the following is the measure of ∠BAC, if AB and AC lie on the opposite sides of the centre?

In the given figure, two chords $AB$ and $AC$ of a circle subtends angles equal to $90^{o}$ and $150^{o}$ , respectively at the centre. Which of the following is the measure of $∠ BAC$ , if $AB$ and $AC$ lie on the opposite sides of the centre?

A
$120^{o}$
B
$240^{o}$
C
$60^{o}$
D
$30^{o}$

Solution:

Consider the given figure,

From the figure-

AB

and

AC

are the chords of a circle with centre

O

∠ AOB = 90^{o}

∠ AOC = 150^{o}

Now,

∠ BOC = 360^{o} - (∠ AOB + ∠ AOC)

∠ BOC = 360^{o} - (90^{o} + 150^{o})

∠ BOC = 120^{o}

. The angle at the centre, subtended by a chord of a circle is double of that of the circumference.
So,

∠ BAC = \frac{1}{2} ∠ BOC

∠ BAC = \frac{1}{2} \times 120^{o}

∠ BAC = 60^{o}

Hence, the measure of

∠ BAC

60^{o} .

Therefore, option 3 is correct.

In the given figure, two chords $AB$ and $AC$ of a circle subtends angles equal to $90^{o}$ and $150^{o}$ , respectively at the centre. Which of the following is the measure of $∠ BAC$ , if $AB$ and $AC$ lie on the opposite sides of the centre?

Solution:

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