AMC 10 · 2008 · #4

Grade 7 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid

circular-sectorangles-around-a-pointarea-circles identify-subproblems ↑ Prerequisites: area-circlesfraction-arithmetic

📏 Short solution 💡 1 insight 📊 Diagram

Problem

Two points on a circle are given by their angles from opposite ends of a diameter. Find what fraction of the circle the smaller sector between them covers.

Pick an answer.

(A)

$\frac {2}{9}$

(B)

$\frac {1}{4}$

(C)

$\frac {5}{18}$

(D)

$\frac {7}{24}$

(E)

$\frac {3}{10}$

How to solve

Strategy Draw a Diagram

The two given angles are measured from two different rays, OA and OB, so they cannot be combined as they stand. Draw the circle and pick one ray, say OA, as a fixed starting line. Then every radius gets a single number: how far you turn from OA to reach it. Once OC and OD carry numbers on the same dial, the angle between them is a subtraction. The condition that C and D are on the same side is what allows one dial to hold both, so it has to be used, not skipped.

1STEP 1

Fix one starting ray

Fixing one ray as zero puts both points on one dial.

OA ↦ 0°, OB ↦ 180°

2STEP 2

Put C and D on the dial

The second angle must be flipped onto that dial.

OC ↦ 30°, OD ↦ 180° - 45° = 135°

3STEP 3

Subtract, then check nothing overlaps

Subtracting gives the angle, and the pieces add back correctly.

∠ COD = 135° - 30° = 105°, 30° + 105° + 45° = 180°

4STEP 4

Which of the two sectors is smaller

That sector really is the smaller of the two.

105° + 255° = 360°, 105 ＜ 255

5STEP 5

Sector area is the angle's share

A sector's area is just the angle's share.

sector/circle = (105/360π r²)/(π r²) = 105/360

6STEP 6

Reduce and read off the choice

Reducing gives 7/24, choice (B).

105/360 = (105 ÷ 15)/(360 ÷ 15) = 7/24

Answer

7/24

Size check first: 105° is a little more than a quarter turn (90°) and clearly less than a third of a turn (120°), so the ratio must land between 1/4 = 0.25 and 1/3 ≈ 0.333. And 7/24 ≈ 0.2917 does. Existence check second, which matters because an angle chase can describe a picture that cannot be drawn: put O at the origin with radius 1, A = (1,0), B = (-1,0), C = (cos 30°, sin 30°), D = (cos 135°, sin 135°). Both C and D have positive height, so they really are on the same side of AB, and the two given angles really are 30° and 45°. So the configuration exists and the answer describes a real picture, not just a consistent set of equations.

💡Key takeaway

Measure every radius from the same starting ray, and the angle you want becomes a subtraction; then a sector is simply that angle's share of 360°.

Fix one starting ray
Put C and D on the dial
Subtract, then check nothing overlaps
Which of the two sectors is smaller
Sector area is the angle's share
Reduce and read off the choice