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

Answer 1

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