AMC 10 · 2025 · #5

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

inscribed-anglearc-measureisosceles-triangle identify-subproblems ↑ Prerequisites: inscribed-angle

📏 Medium solution 💡 3 insights

Problem

In triangle ABC, side AB = 10, side AC = 18, and ∠ B = 130°. The point O is the center of the circle that passes through A, B, and C (the circumcircle). Find the degree measure of ∠ CAO.

Pick an answer.

(A)

$20^\circ$

(B)

$30^\circ$

(C)

$40^\circ$

(D)

$50^\circ$

(E)

$60^\circ$

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The word 'circumcenter' hides three equal radii OA = OB = OC. Drawing the circle and those radii makes the equal lengths visible and reveals two facts to chase: how the 130° angle at B controls the angle at the center O, and how the isosceles triangle AOC then splits into two equal base angles. So the plan is: draw it, break it into the central-angle subproblem and the isosceles-triangle subproblem, and name the unknown base angle to finish.

1STEP 1

Draw the circle and its radii

Draw the circle through A, B, and C, mark its center O, and draw the equal radii OA = OB = OC; ∠ B is inscribed on arc AC.

OA = OB = OC

2STEP 2

Find the central angle AOC

The inscribed 130° doubles to a 260° arc AC, so the angle inside triangle AOC is ∠ AOC = 360° - 260° = 100°.

∠ AOC = 360° - 2(130°) = 360° - 260° = 100°

3STEP 3

Set up the isosceles triangle

In triangle AOC the sides OA and OC are radii, so it is isosceles: ∠ CAO = ∠ ACO = x, and x + x + 100° = 180°.

x + x + 100° = 180°

4STEP 4

Solve for the base angle

180° - 100° leaves 80° for the two equal base angles, so each is 40° — that is choice (C).

2x = 80° → x = 40°, ∠ CAO = 40°

Answer

40°

Add the angles of triangle AOC back up: 40° + 40° + 100° = 180°, which checks out. The result also makes sense of the numbers we were handed: the answer used only ∠ B, and the side lengths 10 and 18 turned out to be a distraction, which is common in this kind of circle problem. 40° is answer (C).

💡Key takeaway

The center of a circle sees an arc twice as wide as the edge does, so a 130° angle at B makes a 100° angle at O, and splitting the leftover in the equal-radius triangle gives 40°.

Draw the circle and its radii
Find the central angle AOC
Set up the isosceles triangle
Solve for the base angle

Draw the circle and its radii

Find the central angle AOC

Set up the isosceles triangle

Solve for the base angle

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