AMC 10 · 2004 · #10

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference
tangent-circlespythagorean-theoremarea-difference identify-subproblemsconvert-to-algebra ↑ Prerequisites: pythagorean-theorem
📏 Short solution 💡 2 insights 📊 Diagram
Problem
Two circles share the same centre, with radii b and c. A line from a point X on the big circle grazes the small circle at Z, and a is that grazing length. Find the area of the ring between the two circles.

Pick an answer.

(A)
$\pi a^2$
(B)
$\pi b^2$
(C)
$\pi c^2$
(D)
$\pi d^2$
(E)
$\pi e^2$
How to solve
Strategy Draw a Diagram

The ring's area is the big circle minus the small circle, so it depends on b squared minus c squared. Reading the figure shows a right triangle hiding in the tangent line, and that right triangle is exactly what turns b squared minus c squared into a single labeled length. Draw the triangle, then let algebra finish.

1STEP 1

Ring equals big minus small

The ring is the big disk minus the small one, so it needs a difference of squares.

Area = π b² - π c² = π(b² - c²)
2STEP 2

Find the right triangle

The grazing line meets the small radius at a right angle, giving a right triangle.

OX² = OZ² + XZ² → b² = c² + a²
3STEP 3

Swap in a single length

That triangle turns the difference into the grazing length, so the ring is π a², choice (A).

b² - c² = a² → Area = π(b² - c²) = π a²
Answer
π a²
The answer pi a squared is the area of a circle with radius a, and it uses only labeled lengths, as required. It correctly ignores d and e, which never entered the ring's area. A quick sanity check: if the inner circle shrinks toward the center (c toward 0), the tangent length a grows toward b, and pi a squared grows toward pi b squared, the full big circle, exactly as a vanishing hole should give.
💡Key takeaway

A tangent line makes a right angle with the radius, so the Pythagorean theorem turns the ring's area into a single circle of radius a.

  • Ring equals big minus small
  • Find the right triangle
  • Swap in a single length

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