The length of arc AXB is twice the length of arc BZC, and

[BTGmoderatorDC]

The length of arc AXB is twice the length of arc BZC, and the length of arc AYC is three times the length of arc AXB. What is the measure of angle BCA?

A. 20
B. 40
C. 60
D. 80
E. 120

OA B

Source: Manhattan Prep

[arosman]

The key to finding arc length is to find the value of the CENTRAL angle. That is because the arc length will always be proportional to the central angle. If you can find the value of an inscribed angle you can always find the value of the central angle and vice versa. Central angle = 2 * inscribed angle.

Start by assigning variables to the given arc lengths:
- BZC = x
- AXB = 2x
- AYC = 6x

This means the entire circle has a circumference of 9x.

Angle BCA is the INSCRIBED angle corresponding to arc AB. Therefore, if we can find the central angle of AB we can then find BCA by dividing by 2.

$$\frac{AXB}{Circuference}$$ = $$\frac{Central\ Angle}{360}$$ ====> $$\frac{2x}{9x}$$ = $$\frac{Central\ Angle}{360}$$

Cross Multiply and solve to get Central Angle = 80

Angle BCA = .5 * 80 = 40

Answer: B