AMC 10 · 2010 · #14

Grade 8 geometry-2d

similar-trianglesequilateral-trianglethirty-sixty-ninety-triangleangle-sum-triangle identify-subproblems ↑ Prerequisites: angle-sum-triangle

📏 Long solution 💡 3 insights

Problem

In triangle ABC the side AB is twice as long as AC. Point D sits on AB and point E sits on BC so that angle BAE equals angle ACD. The segments AE and CD cross at F, and the little triangle CFE turns out to be equilateral. Find the measure of angle ACB.

Pick an answer.

(A)

$60^\circ$

(B)

$75^\circ$

(C)

$90^\circ$

(D)

$105^\circ$

(E)

$120^\circ$

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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

Sketch triangle ABC, mark D, E and F, shade the equilateral triangle CFE, and name the equal angle pair x.

Let x = ∠ BAE = ∠ ACD.

2STEP 2

Angle at F is 120 degrees

A, F, E lie on one line, so angle CFA is 180 minus the equilateral corner angle CFE of 60, leaving 120 degrees.

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

3STEP 3

Triangle ACF fixes angle FAC

In triangle ACF the angle at C is x and the angle at F is 120, so the angle at A is 60 degrees minus x.

∠ FAC = 180° - 120° - x = 60° - x.

4STEP 4

Angle BAC comes out to 60 degrees

Angle BAC splits into angle BAE and angle EAC, so it is x plus 60 minus x — the x cancels and leaves 60 degrees.

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

5STEP 5

Build a helper point M

Let M be the midpoint of AB, so AM = AC with a 60 degree angle between them: triangle AMC is equilateral, giving MC = MB.

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

6STEP 6

Split angle ACB with M

MB = MC makes triangle MBC isosceles, so angle MCB equals angle B and angle ACB is 60 degrees plus angle B.

∠ ACB = ∠ MCA + ∠ MCB = 60° + ∠ B.

7STEP 7

Finish with the triangle angle sum

The three angles sum to 60 plus angle B plus 60 plus angle B = 180, so angle B is 30 and angle ACB is 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