AMC 8 · 1999 · #14

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

perimeterpythagorean-theoremisosceles-trianglearea-rectangles identify-subproblemsarea-difference ↑ Prerequisites: pythagorean-theoremperimeter

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Trapezoid ABCD has AB = CD (isosceles). From the figure, the parallel sides are BC = 8 on top and AD = 16 on the bottom, and the trapezoid's height (perpendicular distance between the parallel sides) is 3. Find the perimeter of ABCD.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The trapezoid's slanted sides are unknown but its height 3 and the two parallel sides 8 and 16 are given. Tool #1 (Draw a Diagram) suggests adding the standard auxiliary lines: drop perpendiculars from B and C down to the bottom base AD. That cut splits the figure into a central rectangle and two congruent right triangles (congruent because the trapezoid is isosceles), which is Tool #7 (Identify Subproblems) — the trapezoid problem becomes a Pythagorean-theorem problem on one easy right triangle.

1STEP 1

Drop perpendiculars from B and C to base AD at E and F: the middle BCFE is a rectangle and the two ends are right triangles.

BE = CF = 3, EF = BC = 8

2STEP 2

The base AD = 16 splits into AE + EF + FD with EF = 8, so the two overhangs satisfy AE + FD = 8.

AE + FD = 16 - 8 = 8

3STEP 3

Isosceles symmetry makes the two triangles congruent, so AE = FD; halving the leftover 8 gives AE = FD = 4.

AE = FD =

\frac{8}{2}

= 4

4STEP 4

Triangle ABE has legs AE = 4 and BE = 3, so the Pythagorean theorem gives the slant AB = 5 (a 3-4-5 triangle).

AB = √(4² + 3²) = √(16 + 9) = √(25) = 5

5STEP 5

The other triangle is congruent, so CD = AB = 5, and the four sides sum to 5 + 8 + 5 + 16 = 34.

Perimeter = AB + BC + CD + DA = 5 + 8 + 5 + 16 = 34 → (D)

Answer

Sanity check the slant length with the triangle inequality and a rough estimate. The slant rises 3 over a run of 4, so it must be longer than 4 but shorter than 4 + 3 = 7 — and 5 sits right in that range. Sanity check the total: the two horizontal sides already contribute 8 + 16 = 24, so the perimeter must exceed 24, which eliminates choices below 24 immediately. Adding two slants of length 5 gives 24 + 10 = 34, matching (D). Choice (E) 48 would require each slant to be 12, far too long for a 3-tall rise; choice (A) 27 would force slants of length 1.5, shorter than the height itself — impossible.

💡Key takeaway

Drop two height lines, split the trapezoid into a rectangle plus two matching right triangles, and a single Pythagorean step on a 3-4-5 triangle gives slant = 5 — perimeter 5 + 8 + 5 + 16 = 34, answer (D).

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