AMC 10 · 2008 · #13

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
area-circlestangent-circlesthirty-sixty-ninety-triangle physical-representationconvert-to-algebra ↑ Prerequisites: area-circles
📏 Medium solution 💡 2 insights
Problem
A small circle sits inside a larger one, touching it from within and touching both sides of a sixty degree angle at the centre. Find the ratio of the two areas.

Pick an answer.

(A)
$\frac{1}{16}$
(B)
$\frac{1}{9}$
(C)
$\frac{1}{8}$
(D)
$\frac{1}{6}$
(E)
$\frac{1}{4}$
How to solve
Strategy Draw a Diagram

The whole problem is about how one circle sits against two rays and inside another circle, so a clean picture is the key. Once it is drawn, two facts fall out: touching both rays forces the small center onto the line that bisects the 60 degree angle, and touching the big circle from inside links the two radii through the distance between centers. Name the two radii, build a right triangle to relate them, then compare their squares to get the area ratio.

1STEP 1

Place the small center on the bisector

Touching both sides puts the centre on the bisector.

∠ AOP = ∠ BOP = 30°
2STEP 2

Relate the radii with a right triangle

A right triangle makes the centre distance twice the small radius.

PT = r, ∠ POT = 30° → OP = 2r
3STEP 3

Use the inside-touch to link the radii

Touching from inside makes that same distance a difference, giving one third.

OP = R - r = 2r → 3r = R → r = R/3
4STEP 4

Compare the areas

Areas go as the square, so the ratio is 1/9, choice (B).

(π r²)/(π R²) = (r/R)² = (1/3)² = 1/9
Answer
1/9
The small radius came out to one third of the big radius, which looks right on the picture: the small circle is clearly much less than half the big one but not tiny. Squaring the one-third gives one ninth, a sensible slice of the area. A quick sanity check on the tangency: with r = R/3, the center sits at OP = 2R/3, and 2R/3 + R/3 = R reaches exactly the far edge of the big circle, confirming the inside touch. So (B) 1/9 is consistent.
💡Key takeaway

Touching both sides of an angle puts a circle on the middle line, and once you know a radius is one third, the area is one ninth because area follows the radius squared.

  • Place the small center on the bisector
  • Relate the radii with a right triangle
  • Use the inside-touch to link the radii
  • Compare the areas

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