AMC 10 · 2018 · #9

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference
similar-trianglesarea-trianglessimilar-figuresratio-proportion identify-subproblems ↑ Prerequisites: similar-trianglesarea-triangles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
Inside isosceles triangle ABC (with AB=AC), a line DE is drawn parallel to base BC, splitting ABC into a top triangle ADE and a bottom trapezoid DBCE. Every triangle in the figure is similar to ABC. A strip of 7 identical smallest triangles sits right above DE, each with area 1, and the whole triangle ABC has area 40. Find the area of trapezoid DBCE.

Pick an answer.

(A)
16
(B)
18
(C)
20
(D)
22
(E)
24

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

Tool #1 (Draw a Diagram): the whole problem is unlocked by reading the figure carefully — spotting that DE is parallel to BC (so △ ADE is a shrunken copy of △ ABC) and that the 4 upward-pointing small triangles line up edge to edge across DE. Tool #7 (Identify Subproblems): instead of measuring the trapezoid directly, split △ ABC into the easy top triangle ADE plus the trapezoid, so trapezoid = big triangle - top triangle. Tool #5 (Look for a Pattern): for similar shapes, area grows as the square of the side ratio, so once we know the side ratio is 4 the area ratio is 4²=16 — no measuring needed.

1STEP 1

Split the big triangle in two

DE cuts △ ABC into the top triangle ADE and the trapezoid DBCE, so the trapezoid is 40 minus △ ADE.

[DBCE] = [ABC] - [ADE] = 40 - [ADE]
2STEP 2

ADE is a scaled-down ABC

Since DE ∥ BC, △ ADE has the same angles as △ ABC, so △ ADE, △ ABC, and each small triangle are all similar.

△ ADE ∼ △ ABC ∼ △(small)
3STEP 3

Find the side ratio along DE

The 4 upward triangles tile DE end to end, so DE is 4 small bases long and △ ADE is the small triangle scaled up by 4.

DE = 4 × (small base) → side ratio = 4
4STEP 4

Square the ratio to get the area

For similar shapes area scales as the square of length, so △ ADE = 4² times one small triangle, giving area 16.

[ADE] = 4² × 1 = 16
5STEP 5

Subtract to finish

Put it together: the trapezoid is the whole triangle minus the top triangle, 40 - 16 = 24.

[DBCE] = 40 - 16 = 24 → (E)
Answer
24
The trapezoid is the bigger, wider bottom slab of the triangle, so it should hold most of the area — 24 out of 40 is more than half, which fits. A cleaner cross-check: triangle ADE of side ratio 4 is built from 16 smallest triangles arranged in rows of 7,5,3,1 from bottom to top (7+5+3+1=16), so [ADE]=16 and [DBCE]=40-16=24. The smaller choices (A) 16 through (D) 22 would each force △ ADE to be bigger than 16, but the side ratio of 4 pins [ADE] at exactly 16, so (E) is the only consistent value.
💡Key takeaway

A line parallel to the base trims off a smaller copy of the triangle; if its side is 4 times smaller, its area is 4²=16 times smaller, so the leftover trapezoid is 40-16=24.

  • Split the big triangle in two
  • ADE is a scaled-down ABC
  • Find the side ratio along DE
  • Square the ratio to get the area
  • Subtract to finish

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