AMC 10 · 2010 · #8

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
similar-trianglesequilateral-trianglethirty-sixty-ninety-triangleangle-sum-triangle identify-subproblems ↑ Prerequisites: angle-sum-triangle
📏 Long solution 💡 3 insights
Problem
Two marked angles in a triangle are equal and a small triangle formed by the crossing segments is equilateral. Find the angle at C.

Pick an answer.

(A)
$60^\circ$
(B)
$75^\circ$
(C)
$90^\circ$
(D)
$105^\circ$
(E)
$120^\circ$
How to solve
Strategy Draw a Diagram

The problem is packed with points and equal angles, so the first move is a clean diagram to see how everything connects at F. Then name the shared angle with a variable and chase angles piece by piece: the equilateral triangle pins the angle at F, one small triangle fixes angle A, and the length rule AB = 2*AC finishes the job.

1STEP 1

Draw and label the figure

One letter names both equal angles.

Let x = ∠ BAE = ∠ ACD.
2STEP 2

Angle at F is 120 degrees

The crossing angle is 120 degrees.

∠ CFA = 180° - ∠ CFE = 180° - 60° = 120°.
3STEP 3

Triangle ACF fixes angle FAC

The small triangle fixes one part of angle A.

∠ FAC = 180° - 120° - x = 60° - x.
4STEP 4

Angle BAC comes out to 60 degrees

Adding the parts makes angle A exactly 60 degrees.

∠ BAC = ∠ BAE + ∠ EAC = x + (60° - x) = 60°.
5STEP 5

Build a helper point M

A midpoint on the doubled side builds an equilateral triangle.

AM = 1/2AB = AC, ∠ A = 60° → △ AMC equilateral, MA = MB = MC.
6STEP 6

Split angle ACB with M

That point splits angle C into two known pieces.

∠ ACB = ∠ MCA + ∠ MCB = 60° + ∠ B.
7STEP 7

Finish with the triangle angle sum

The angle sum finishes it at 90 degrees.

60° + ∠ B + (60° + ∠ B) = 180° → ∠ B = 30°, ∠ ACB = 60° + 30° = 90°. (C)
Answer
90°
Check the triangle we found: angles 60, 30, 90 with the right angle at C. Then AB is the hypotenuse and AC is the side opposite the 30 degree angle, which is always half the hypotenuse, so AB = 2*AC exactly as required. Everything fits, so angle ACB = 90 degrees is consistent.
💡Key takeaway

Name the mystery angle, chase the angles around the equilateral piece until it cancels, then let the side rule AB = 2*AC snap the last angle into place.

  • Draw and label the figure
  • Angle at F is 120 degrees
  • Triangle ACF fixes angle FAC
  • Angle BAC comes out to 60 degrees
  • Build a helper point M
  • Split angle ACB with M
  • Finish with the triangle angle sum

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