Problem 15 · 2014 AMC 8
Medium
Geometry & Measurement
central-angleisosceles-triangle
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?
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Answer: C — 90 degrees.
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Hint 1 of 2
The 12 equal arcs turn the picture into a clock: each arc is 360°÷12 = 30°. Counting arcs gives you every central angle for free.
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Hint 2 of 2
Each marked triangle has two radii as sides, so it's isosceles — once you know the apex (central) angle, the two base angles are each (180° − apex)/2.
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Approach: central angles from arc-counting, then isosceles base angles
- x sits in ▵OAE. Arc A→E spans 4 arcs, so the central angle ∠AOE = 4 × 30° = 120°. Two radii make it isosceles, so x = (180° − 120°)/2 = 30°.
- y sits in ▵OIG. Arc G→I spans 2 arcs, so ∠GOI = 60° and y = (180° − 60°)/2 = 60°.
- x + y = 30° + 60° = 90°.
Another way — add the central angles first, then halve:
- Both x and y are base angles of isosceles triangles, so each equals (180° − its central angle)/2.
- Adding: x + y = [(180° − 120°) + (180° − 60°)]/2 = (60° + 120°)/2 = 180°/2 = 90°.
- Spotting that the two leftover arc-angles (120° and 60°) total 180° collapses the work to one division.
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