AMC 8 · 2014 · #15

Grade 8 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

inscribed-anglearc-measureequal-spacing identify-subproblems ↑ Prerequisites: angle-sum-triangleequal-spacing

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A circle with center O has its circumference cut into 12 equal arcs, with the breakpoints labeled A, B, C, …, L in order. Inside the figure, angle x is the angle at vertex A in the path E–A–G (so x = ∠ EAG), and angle y is the angle at vertex G in the path A–G–I (so y = ∠ AGI). Find x + y in degrees.

Pick an answer.

(A)

(B)

(C)

(D)

120

(E)

150

AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The figure is busy, but the question is just x + y, so Tool #7 (Identify Subproblems) splits the work into three clean sub-questions: (a) what is the measure of one small arc, (b) what is the value of x, (c) what is the value of y. Tool #1 (Draw a Diagram) is the supporting move — adding the radii OE, OG, OA, OI to the figure turns each unknown angle on the circle into two isosceles triangles whose equal sides are radii. With those isosceles triangles in view, we can find x and y from the central angles using only the triangle-angle-sum and the straight-line (180°) fact, never needing the inscribed-angle theorem as a black box.

1STEP 1

The 12 equal arcs fill 360°, so one small arc — the central angle at O — measures $\frac{360°}{12}$ = 30°.

one small arc =

\frac{360°}{12}

= 30°

2STEP 2

Draw radii OE, OA, OG; arc E→G spans 2 small arcs, so ∠ EOG = 60°, and OA = OE = OG makes the radius triangles isosceles.

∠ EOG = 2 · 30° = 60°; OA = OE = OG

3STEP 3

The radius triangles are isosceles, and the angle-sum bookkeeping shows x is half of ∠ EOG: x = 30°.

x =

\frac{1}{2}

· ∠ EOG =

\frac{1}{2}

· 60° = 30°

4STEP 4

Repeat for y = ∠ AGI: the minor arc A→I is 4 small arcs, so ∠ AOI = 120° and y = 60°.

∠ AOI = 4 · 30° = 120°; y =

\frac{1}{2}

· 120° = 60°

5STEP 5

Add the parts: x + y = 30° + 60° = 90° — choice (C).

x + y = 30° + 60° = 90° → (C)

Answer

Each small arc is 30°, x sits on a 2-arc span, and y sits on a 4-arc span — a total span of 6 arcs = 180° on the circle. The corresponding on-circle (inscribed) angle sum is exactly half of that, 90°, matching choice (C). Visually, x and y in the figure look acute and roughly 30° and 60°, which fits.

💡Key takeaway

When an angle's vertex sits on a circle, the radii drawn to its sides make isosceles triangles — and the triangle-angle-sum trick turns the angle into half of the arc it spans.

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