Competition · AMC preparation · step 4 of 4

AMC 8 · 2023 · #19

Grade 7 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference

area-trianglessimilar-figuresratio-proportion area-differenceeasier-related-problem ↑ Prerequisites: area-trianglesratio-proportion

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A smaller equilateral triangle is placed inside a larger equilateral triangle so that the ring-shaped region between them splits into three congruent trapezoids. The smaller triangle's side is

\frac{2}{3}

of the larger triangle's side. Find the ratio of one trapezoid's area to the inner triangle's area.

Pick an answer.

(A)

1 : 3

(B)

3 : 8

(C)

5 : 12

(D)

7 : 16

(E)

4 : 9

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The ratio asked for is the same no matter what concrete side length we use, so we replace the abstract setup with the easiest case: outer side = 3 units and inner side = 2 units. Because both triangles are equilateral, the inner-to-outer area ratio must be (2/3)² = 4/9, so we can simply name the inner area 4 square-units and the outer area 9 square-units — no square-root formula needed. We then split the work into two subproblems (Tool 7): (i) find the combined area of the three trapezoids, and (ii) divide by 3. The combined-trapezoid area is found by looking at the complement (Tool 16): outer minus inner. A quick diagram (Tool 1) makes the partition concrete.

1STEP 1

Sketch the nested triangles

The inner triangle plus the three trapezoids tile the outer triangle, so the trapezoids fill the outer area minus the inner area.

3 · T = A_outer - A_inner

2STEP 2

Pick easy side lengths

Both triangles are equilateral, so they are similar with scale factor $\frac{2}{3}$ ; areas scale by the square, so inner : outer = $\frac{4}{9}$ .

A_inner/A_outer = (2/3)² = 4/9

3STEP 3

Pick convenient areas

Pick units so the areas are whole numbers: inner area = 4, outer area = 9. This keeps the 4 : 9 ratio and dodges any square roots.

A_inner = 4, A_outer = 9

4STEP 4

Subtract to get one trapezoid

The three congruent trapezoids fill the ring, so their total is 9 - 4 = 5; dividing by 3 gives one trapezoid = $\frac{5}{3}$ .

3T = 9 - 4 = 5 ⟹ T = 5/3

5STEP 5

Form and simplify the ratio

Form the asked ratio: ( $\frac{5}{3}$ ) : 4 = 5 : 12, the trapezoid-to-inner-triangle ratio.

T/A_inner = 5/3/4 = 5/12 ⟹ T : A_inner = 5 : 12

Answer

5 : 12

Sanity-check the totals: one trapezoid has area

\frac{5}{3}

, so three trapezoids together have area 5, and adding the inner triangle's area 4 gives 9, matching the outer triangle's area. The ratio 5{:}12 is also between

\frac{1}{3}

(≈ 0.333) and

\frac{4}{9}

(≈ 0.444), which are the surrounding answer choices, so the magnitude is reasonable for a 'one of three trapezoids' slice. Choices (A) 1{:}3 would mean three trapezoids equal the inner triangle (i.e. ring = inner), but the ring should be larger than the inner since 9 - 4 = 5 ＞ 4, so (A) is too small; (E) 4{:}9 matches the inner-to-outer ratio, not the requested one, so it is the classic distractor.

💡Key takeaway

This AMC 8 problem only needs Grade 7 'scale-drawing area ratio' (sides scaled by k make areas scaled by k²) you already know!

Sketch the nested triangles
Pick easy side lengths
Pick convenient areas
Subtract to get one trapezoid
Form and simplify the ratio

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Sketch the nested triangles

Pick easy side lengths

Pick convenient areas

Subtract to get one trapezoid

Form and simplify the ratio

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