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In the figure, C is a point on the minor arc AB of the circle, with centre O. Given $∠ A C B = x ∘$ and $∠ A O B = y ∘$ express $y$ in terms of $x$ . Calculate $x$ , if ACBO is a parallelogram.

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An Intiative by Sri Chaitanya

x = 1200

x = 1500

x = 1800

x = 1100

answer is A.

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Detailed Solution

Here in this problem, the major arc AB subtends a central angle at the center O with its radii OA, OB as a reflex

, whose measure we denoted in the diagram as . So

The great arc AB here also subtends the inscribed angle at C on the minor arc AB as

with its chords AC, BC, the measure of which is given in the question as

From the inscribed angle theorem, we know that the central angle is twice the inscribed angle. so we have

reflex ∠ A O B = 2 ∠ A C B ⇒ z ° = 2 x °

We are also given in the question the obtuse

∠ A O B = y ∘

. We see that obtuse

∠ A O B

and reflex

∠ A O B

make the complete angle of measure

360 °

. So we have

z ∘ + y ∘ = 360 ∘ ⇒ y ∘ = 360 ∘ − z ∘ ⇒ y ∘ = 360 ∘ − 2 x ∘

The expression of y in terms of x is

. Next, the question asks us to calculate x if ACBO is a parallelogram. We know that opposite angles in a parallelogram are equal. so we have

∠ A C B = ∠ A O B ⇒ x ° = y °

We put previously obtained

y ∘ = 360 ∘ − 2 x ∘

and proceed to get

x °

as,

⇒ x ∘ = 360 − 2 x ∘ ⇒ 3 x ° = 360 ° ⇒ x = 120 °

So, the correct answer is Option 1.

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