Question

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

Answer 1

[Error in Question : The ratio of arc ABC to the circumference of the circle is to be derived here, not PQR]

Let r be the radius of the circle.

Given, ABC is the arc of circle.

Construction, 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 = 1/4 × 2πr
∴ Length of arc ABC = 1/4 × Circumference of the Circle
Thus, Length of arc ABC : Circumference of the circle = 1 : 4

Answer 2

In the Question it is written arc PQR but it is arc ABC

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 = 135⁰              (Given)

∠ABC + ∠ADC = 180⁰  (Sum of opposite angles of a cyclic quadrilateral is 180⁰)

∴ 135⁰ + ∠ADC = 180⁰

⇒ ∠ADC = 180⁰ - 135⁰ = 45⁰

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 × 45⁰ = 90⁰

Thus, the arc ABC represent quadrant of the circle.

Length of arc ABC =
                               1/4 × 2πr

∴ Length of arc ABC =

⇒ Length of arc ABC : Circumference of the circle

=    1 : 4