Question

The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y?$

$75$

$80$

$90$

$120$

$150$

Solution:

Note that each of the $12$ arcs splits the circle evenly, so they each cover $360^{\circ} / 12 = 30^{\circ}.$

$\angle AOE$ spans $4$ of these arcs, so $\angle AOE = 4 \cdot 30^{\circ} = 120^{\circ}.$ Similarly, $\angle GOI = 2 \cdot 30^{\circ} = 60^{\circ}.$ We also know that both triangles are isosceles since two of their sides are radii. Therefore, $x = \dfrac{180 - 120}{2} = 30^{\circ}$ and $y = \dfrac{180 - 60}{2} = 60^{\circ}.$ Therefore, $x + y = 90^{\circ}.$

Thus, C is the correct answer.

2014 AMC 8 Problem 15

Solution:

Problem 15 in Other Years