if ABC is an arc of a circle and angleABC=135,then the ratio of arc PQR to the circumference is

Let r be the radius of the circle.

Given, ABC is the arc of circle.

Take a point D in the alternative segment. Join AD and CD.

∠ABC = 135°         (Given)

∠ABC + ∠ADC = 180°       (Sum of opposite angles of a cyclic quadrilateral is 180°)

∴ 135° + ∠ADC = 180°

⇒ ∠ADC = 180° – 135° = 45°

Now, ∠AOC = 2 × ∠ADC           (The angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle)

∴ ∠AOC = 2 × 45° = 90°

Thus, the arc ABC represent quadrant of the circle.

Length of arc ABC =

∴ Length of arc ABC =

⇒ Length of arc ABC : Circumference of the circle = 1 : 4