Circle O (see below) has center O, diameter AB and a radius of 6. Line CD is parallel to the diameter.
What is the perimeter of the shaded region?
The perimeter will be the length of the arc CAE and the line segments CB and EB.
Calculate arc CAE
Step 1: Angle X = 30 degrees, because they are alternate interior angles
Step 2: Angle CBE = 2X = 2*30 degrees = 60 degrees
Step 3: Angle COE = 120 degrees because COE is a corresponding angle of the inscribed angle CBE. Corresponding angles = 2x inscribed angle’s measure
Step 4: arc = (120 degrees / 360 degrees) * circumference = 1/3 * 2(radius)pi = 1/3 * (2)(6)pi = 4pi
Calculate lines CB and EB
Step 1: Triangle ACB is a 30-60-90 triangle because one it’s in a semicircle. Angle C = 90 degrees.
Step 2: 30-60-90 triangle = x – rad3(x) – 2x
Step 3: line segment CB is opposite 60 degrees, so the length is radical3(x) = radical3(6)
Step 4: line segment EB is the same as CB, so EB = radical3(6)
Add them together:
perimeter of shaded region = 4pi + radical3(6) + radical3(6)
perimeter of shaded region = 4pi + 12 radical3
*after the timed practice test & looked at my notes, but still. I RULE (for one, brief, shining moment).