AMC 10 · 2025 · #10

Grade 10 geometry-2d
Archetype Path Length Comparison & Shortest Path └ Sub-archetype Sum Known Path Segments
arc-length-radiansystems-of-equationsquadratic-equations convert-to-algebra ↑ Prerequisites: circle-circumference
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
Two arcs share the same center O. The inner arc is the major arc AD; the outer arc is the minor arc BC. A sits between O and B on one straight radius, and D sits between O and C on another straight radius. The major arc AD, the minor arc BC, and both straight segments AB and CD each have length 2 pi. Find the distance from O to A.

Pick an answer.

(A)
1
(B)
$1 - \pi + \sqrt{\pi^{2} + 1}$
(C)
$\frac{\pi}{2}$
(D)
$\frac{\sqrt{\pi^{2} + 1}}{2}$
(E)
2
How to solve
Strategy Introduce a Variable

Every arc length pins together a radius and a central angle, but the angle is never given. Naming that angle as a second variable turns the two arc-length facts into two equations in the same two unknowns. With two equations and two unknowns, the angle can be eliminated and the radius solved for exactly, which is why introducing the hidden angle is the move that unlocks the whole problem.

1STEP 1

Read the two radii off the figure

A lies on OB with AB = 2 pi, so OB = r + 2 pi (OC too). The minor arc BC uses the opening angle; the major arc AD takes the rest.

OA = OD = r, OB = OC = r + 2π
2STEP 2

Name the angle and write both arc lengths

Name theta the minor arc's angle. Arc = radius times angle gives (1) r(2 pi - theta) = 2 pi and (2) (r + 2 pi) theta = 2 pi.

(1) r(2π - θ) = 2π (2) (r + 2π) θ = 2π
3STEP 3

Add the equations to kill the angle

Adding (1) and (2) cancels the r*theta cross terms, leaving 2 pi r + 2 pi theta = 4 pi. Divide by 2 pi: r + theta = 2, so theta = 2 - r.

r(2π-θ) + (r+2π)θ = 4π → 2π r + 2πθ = 4π → r + θ = 2
4STEP 4

Substitute back to get a quadratic

Put theta = 2 - r into (1): r(r + 2 pi - 2) = 2 pi, i.e. r^2 + (2 pi - 2) r - 2 pi = 0. Solve with a = 1, b = 2 pi - 2, c = -2 pi.

r(r + 2π - 2) = 2π → r² + (2π - 2)r - 2π = 0
5STEP 5

Simplify the root and pick the valid one

The discriminant 4 pi^2 + 4 = 4(pi^2 + 1) has root 2 sqrt(pi^2 + 1). A radius must be positive, so only the plus sign survives — choice (B).

r = (-(2π-2) ± 2√(π²+1))/2 = 1 - π ± √(π² + 1) → r = 1 - π + √(π² + 1)
Answer
1 - π + √(π² + 1)
Numerically r = 1 - pi + sqrt(pi² + 1) is about 1.155, so theta = 2 - r is about 0.845 radians and the outer radius r + 2 pi is about 7.438. Check the outer arc: 7.438 times 0.845 is about 6.28, which is 2 pi. Check the inner major arc: 1.155 times (2 pi - 0.845) = 1.155 times 5.438 is about 6.28, again 2 pi. Both arc conditions hold, so the value is consistent. It is also positive and smaller than the outer radius, as a inner radius should be.
💡Key takeaway

When an arc length hides its angle, give the angle a name so each arc becomes an equation, then combine the equations to make the angle disappear.

  • Read the two radii off the figure
  • Name the angle and write both arc lengths
  • Add the equations to kill the angle
  • Substitute back to get a quadratic
  • Simplify the root and pick the valid one

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