AMC 10 · 2014 · #15

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
thirty-sixty-ninety-trianglearea-trianglesarea-rectangles identify-subproblems ↑ Prerequisites: thirty-sixty-ninety-triangle
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
In rectangle ABCD the bottom side DC is twice the vertical side CB. Two rays from corner D, namely DE and DF, cut the right angle ∠ ADC into three equal pieces, and E and F land on the top side AB. Find how the area of triangle DEF compares to the area of the whole rectangle.

Pick an answer.

(A)
$\ \frac{\sqrt{3}}{6}$
(B)
$\frac{\sqrt{6}}{8}$
(C)
$\frac{3\sqrt{3}}{16}$
(D)
$\frac{1}{3}$
(E)
$\frac{\sqrt{2}}{4}$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

This is a positions-and-shapes problem, so tool #1 (Draw a Diagram) leads: put D at the origin with DC along the x-axis, which turns the trisected right angle into clean 30° rays and makes E and F easy to pin down. Because only a ratio of areas is asked, tool #4 (Introduce a Variable) lets us fix the short side CB=1 once and for all. Then tool #7 (Identify Subproblems) splits the work into three bites: locate E and F on the top edge, find the area of △ DEF, and divide by the rectangle's area.

1STEP 1

Set coordinates and split the right angle

Put D at the origin and take CB = 1, so DC = 2; the right angle at D splits into three slices of 30° each.

∠ ADC = 90° = 30° + 30° + 30°, CB = 1, DC = 2
2STEP 2

Locate E and F on the top edge

Triangles DAE and DAF are 30-60-90 with leg DA = 1, so AE = √(3)/3 and AF = √(3).

AE = 1/√(3) = √(3)/3, AF = √(3)
3STEP 3

Area of triangle DEF

Base EF = AF - AE = 2√(3)/3 lies flat on the top edge and the height is 1, so [△ DEF] = √(3)/3.

EF = √(3) - √(3)/3 = 2√(3)/3, [△ DEF] = 1/2·2√(3)/3 · 1 = √(3)/3
4STEP 4

Compare to the rectangle and pick the choice

The rectangle's area is 2 · 1 = 2, so the ratio is (√(3)/3)/2 = √(3)/6 — choice (A), and no other choice matches.

([△ DEF])/[ABCD] = (√(3)/3)/2 = √(3)/6 → (A)
Answer
√(3)/6
The number √(3)/6≈ 0.29 is a believable fraction of the rectangle: the triangle spans from E at x=√3/3≈0.58 to F at x=√3≈1.73 across a rectangle that runs from 0 to 2, so it covers a bit more than a quarter — consistent with 0.29. The answer also stays the same no matter what value we pick for CB, since choosing CB=1 was a free scaling choice for a ratio. Note this leans on the 30-60-90 side ratios, which is early high-school geometry sitting at the top of the K-8 range.
💡Key takeaway

Anchor the corner at the origin, use the fixed 1:√(3):2 ratios of a 30-60-90 triangle to place the points, then compare the triangle's area to the rectangle's.

  • Set coordinates and split the right angle
  • Locate E and F on the top edge
  • Area of triangle DEF
  • Compare to the rectangle and pick the choice

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