AMC 10 · 2013 · #9

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
similar-trianglesisosceles-triangleperimeter identify-subproblems ↑ Prerequisites: similar-triangles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
Two segments drawn parallel to the equal sides cut a parallelogram out of a triangle. Find its perimeter.

Pick an answer.

(A)
48
(B)
52
(C)
56
(D)
60
(E)
72
How to solve
Strategy Draw a Diagram

The figure is the key. Working from the marked diagram (Tool #1), the perimeter of parallelogram ADEF is AD+DE+EF+FA, and opposite sides are equal, so it collapses to 2(AD+AF). The parallel line DE ∥ AC cuts a small triangle BDE off the corner at B; reading the equal angles straight off the figure shows that small triangle is isosceles, which pins down one of the lengths. To see that the exact spot of D does not matter, name BD=x (Tool #4) and watch it cancel. Breaking the perimeter into the little triangle plus the parallelogram is Tool #7, Identify Subproblems.

1STEP 1

Write the perimeter from the figure

The perimeter is twice a sum of two sides.

P = AD + DE + EF + FA = 2 (AD + AF)
2STEP 2

Spot the isosceles corner triangle

The parallel makes a corner triangle isosceles.

∠ DBE = ∠ B = ∠ C = ∠ DEB → BD = DE
3STEP 3

Name BD and let it cancel

Naming one segment lets it cancel.

AD + AF = (28 - x) + x = 28
4STEP 4

Double it for the perimeter

The perimeter is 56, choice (C).

P = 2(AD + AF) = 2 · 28 = 56 → (C)
Answer
56
The parallelogram lives inside the triangle, so its perimeter should be less than the triangle's perimeter 28+28+20 = 76; indeed 56 ＜ 76. The result 2 · AB = 56 uses only the equal side 28 and ignores BC=20, which fits the fact that x cancelled: the answer cannot depend on where D sits. Every answer choice is even, matching P = 2(AD+AF), and 56 is on the list as (C).
💡Key takeaway

A line drawn parallel to a side of an isosceles triangle cuts off another isosceles triangle, and the parallelogram's two sides always add back to the full slant side, so the perimeter is just twice that side.

  • Write the perimeter from the figure
  • Spot the isosceles corner triangle
  • Name BD and let it cancel
  • Double it for the perimeter

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