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Question
If ABC is an arc of a circle and ∠ABC = 135°, then the ratio of arc \[\stackrel\frown{ABC}\] to the circumference is ______.
Options
1 : 4
3 : 4
3 : 8
1 : 2
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Solution
If ABC is an arc of a circle and ∠ABC = 135°, then the ratio of arc \[\stackrel\frown{ABC}\] to the circumference is 1 : 4.
Explanation:
Given:
ABC is an arc.
∠ABC = 135°,
Calculation:
Let r be the radius of the circle.
Construction, take a point D in the alternative segment. Join AD to CD.
∠ABC + ∠ADC = 180° (opposite angles of a cyclic quadrilateral)
= 135° + ∠ADC = 180°
= ∠ADC = 180° − 135°
= ∠ADC = 45°
∠AOC = 2 × ADC
= ∠AOC = 2 × 45° = 90°
Hence, the arc ABC represent quadrant of the circle.
Length of arc ABC = `[1/4]` × 2πr
= Length of arc ABC = `[1/4]` × Circumference of the Circle
Length of arc ABC : Circumference of the circle
= 1 : 4
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