AMC 8 · 2019 · #4

Grade 8 geometry-2d

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

perimeterpythagorean-theoremarea-triangles identify-subproblemsarea-difference ↑ Prerequisites: pythagorean-theoremperimeterarea-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Rhombus ABCD has perimeter 52 meters and one diagonal AC of length 24 meters. Find the area of the rhombus in square meters.

Pick an answer.

(A)

(B)

(C)

105

(D)

120

(E)

144

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The problem is a 2-D geometry question, so Tool #1 (Draw a Diagram) is the natural entry point: sketch the rhombus, add both diagonals, and mark where they cross. Once the diagonals are drawn, Tool #7 (Identify Subproblems) makes the path obvious — the rhombus splits into four congruent right triangles, so the problem decomposes into (i) get the side length from the perimeter, (ii) get the missing half-diagonal from a right triangle, (iii) combine the two diagonals into the area formula.

1STEP 1

A rhombus has four equal sides, so dividing the perimeter by 4 gives one side of 13 meters.

side =

\frac{52}{4}

= 13 meters

2STEP 2

Both diagonals split the rhombus into four right triangles meeting at center M, where AM is half of AC = 12 meters.

AM =

\frac{AC}{2}

\frac{24}{2}

= 12 meters

3STEP 3

In right triangle AMB the side AB=13 is the hypotenuse and AM=12 a leg, so the Pythagorean theorem gives BM = 5.

AM² + BM² = AB² → 12² + BM² = 13² → BM² = 169 - 144 = 25 → BM = 5

4STEP 4

Since the diagonals bisect each other, double BM to get the full second diagonal BD = 10 meters.

BD = 2 × BM = 2 × 5 = 10 meters

5STEP 5

The rhombus area is half the product of the diagonals: $\frac{1}{2}$ × 24 × 10 = 120 square meters.

Area =

\frac{1}{2}

× d₁ × d₂ =

\frac{1}{2}

× 24 × 10 = 120 square meters → (D)

Answer

120

Quick sanity check: the rhombus fits inside the 24 × 10 rectangle formed by its diagonals, which has area 240. The rhombus is exactly half of that rectangle (its four triangle pieces fill half), giving 120 square meters. The answer also lies between 60 (a very 'flat' rhombus) and 144 (the would-be square with side √(144)=12), so 120 is in the right magnitude. Choice (D) is confirmed.

💡Key takeaway

This AMC 8 problem only needs the Grade 8 Pythagorean theorem (the 5-12-13 right triangle hiding inside a rhombus) you already know!

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