AMC 10 · 2007 · #4

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
isosceles-triangleangle-sum-triangleinscribed-angle identify-subproblems ↑ Prerequisites: angle-sum-triangle
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
O is the center of the circle that passes through all three vertices of △ ABC. Two of the central angles are known: ∠ BOC=120° and ∠ AOB=140°. Find the measure of the triangle's angle at B, ∠ ABC.

Pick an answer.

(A)
35
(B)
40
(C)
45
(D)
50
(E)
60

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The center O is joined to all three vertices, so the picture is really three triangles △ AOB, △ BOC, △ AOC that share the point O. Tool #1 (Draw a Diagram) matters because the key fact is only visible once you mark that OA, OB, OC are equal radii — that makes each of the three triangles isosceles. Tool #7 (Identify Subproblems) then handles the angle at B by splitting it with OB into two base angles, one from each neighbouring isosceles triangle, which we can find separately and add.

1STEP 1

Find the third central angle

The three central angles at O fill one full turn, so ∠ AOC=360°-140°-120°=100°.

∠ AOC=360°-∠ AOB-∠ BOC=360°-140°-120°=100°
2STEP 2

Base angles of triangle AOB

OA=OB are radii, so △ AOB is isosceles and its base angles match: ∠ OBA=(180°-140°)/2=20°.

∠ OBA=(180°-∠ AOB)/2=(180°-140°)/2=20°
3STEP 3

Base angles of triangle BOC

Likewise OB=OC, so △ BOC is isosceles with apex 120°: ∠ OBC=(180°-120°)/2=30°.

∠ OBC=(180°-∠ BOC)/2=(180°-120°)/2=30°
4STEP 4

Add the two pieces at B

OB splits ∠ ABC into those two base angles, so ∠ ABC=20°+30°=50°, choice (D).

∠ ABC=∠ OBA+∠ OBC=20°+30°=50° → (D)
Answer
50
Each base angle came from a valid isosceles triangle: 20°+20°+140°=180° and 30°+30°+120°=180°, both check out. The answer 50° is a plausible triangle angle (between 0° and 180°) and matches choice (D). As a full consistency test, the other two triangle angles come out as ∠ BAC=(180-100)/2+20-type sums; more simply, each interior angle of △ ABC is half of the central angle across from it, giving 120/2=60° at A, 100/2=50° at B, 140/2=70° at C, and 60+50+70=180°.
💡Key takeaway

Draw the radii from the center: every radius has the same length, so each triangle is isosceles, and their equal base angles let you build the angle you want by adding pieces.

  • Find the third central angle
  • Base angles of triangle AOB
  • Base angles of triangle BOC
  • Add the two pieces at B

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