Find the length of the arc shown in red, leave your answer in terms of pi.

The red arc has length (3/2) π

Step-by-step explanation:

* Lets revise some rules in the circle

- The measure of any arc = the measure of the central angle

  subtended by this arc

- Equal central angles subtended by equal arcs

- The length of the arc is a part from the length of the circle,

 it depends on the ratio between the measure of this arc

 and the measure of the circle

- The measure of the circle = 360°

- The length of the circle = 2πr

* Lets solve our problem

∵ The measure of the arc = 30°

∴ The measure of its central angle 30°

∴ The central angle of the red arc = 30° by condition of

   vertically opposite angles are equal in measures

∴ The measure of the red arc = 30°

- The ratio between measure the arc and the measure

  of the circle is 30/360 = 1/12

∴ The arc is 1/12 from the circle

∴ The length of the arc is 1/12 from the length of the circle

∴ Length red arc = (1/12) × 2πr

∵ r = 9

∴ Length red arc = (1/12) × 2 × π × 9 = (3/2) π

* The red arc has length (3/2) π

example 2 – simplify: step 1: completely factor both the numerators and denominators of all fractions. step 2: change the division sign to a multiplication sign and flip (or reciprocate) the fraction after the division sign; essential you need to multiply by the reciprocal. step 3 : cancel or reduce the fractions.