AMC 8 · 2019 · #4

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference

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

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$ ?

$\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144$

Pick an answer.

(A)

(B)

(C)

105

(D)

120

(E)

144

View mode:

Toolkit + CCSS Solution

Understand

Restated: Rhombus $ABCD$ has perimeter $52$ meters and one diagonal $\overline{AC}$ of length $24$ meters. Find the area of the rhombus in square meters.

Givens: $ABCD$ is a rhombus (all four sides equal); Perimeter $= 52$ meters; Diagonal $AC = 24$ meters; Answer choices: (A) $60$, (B) $90$, (C) $105$, (D) $120$, (E) $144$

Unknowns: Area of rhombus $ABCD$ in square meters

Understand

Restated: Rhombus $ABCD$ has perimeter $52$ meters and one diagonal $\overline{AC}$ of length $24$ meters. Find the area of the rhombus in square meters.

Givens: $ABCD$ is a rhombus (all four sides equal); Perimeter $= 52$ meters; Diagonal $AC = 24$ meters; Answer choices: (A) $60$, (B) $90$, (C) $105$, (D) $120$, (E) $144$

Plan

Primary tool: #1 Draw a Diagram

Secondary: #7 Identify Subproblems

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.

Execute — Answer: D

#1 Draw a Diagram 3.OA.A.3 Step 1

Find the side length.
A rhombus has four equal sides, so dividing the perimeter by $4$ gives one side.

$$\text{side} = \dfrac{52}{4} = 13 \text{ meters}$$

💡 Splitting the total perimeter evenly among $4$ equal sides is a Grade 3 multiplication/division word-problem move.

#1 Draw a Diagram 5.G.B.4 Step 2

Sketch the rhombus with both diagonals.
Because the diagonals of a rhombus are perpendicular and bisect each other, they meet at the center $M$ and cut the rhombus into four congruent right triangles.
Half of diagonal $AC$ is one leg: $AM = 24 / 2 = 12$ meters.

$$AM = \dfrac{AC}{2} = \dfrac{24}{2} = 12 \text{ meters}$$

💡 Recognizing the rhombus's diagonals as perpendicular bisectors is part of classifying special quadrilaterals in Grade 5.

#7 Identify Subproblems 8.G.B.7 Step 3

Zoom in on right triangle $\triangle AMB$.
The hypotenuse is the rhombus's side ($AB = 13$), one leg is $AM = 12$, and the other leg $BM$ is the unknown half of the second diagonal.
Apply the Pythagorean theorem to solve for $BM$.

$$AM^2 + BM^2 = AB^2 \;\Rightarrow\; 12^2 + BM^2 = 13^2 \;\Rightarrow\; BM^2 = 169 - 144 = 25 \;\Rightarrow\; BM = 5$$

💡 Splitting the rhombus produces a right triangle with two known sides — exactly the Grade 8 Pythagorean-theorem setup.

#7 Identify Subproblems 5.G.B.4 Step 4

Double $BM$ to get the full second diagonal $BD$, since the diagonals bisect each other.

$$BD = 2 \times BM = 2 \times 5 = 10 \text{ meters}$$

💡 Using the bisecting property of the rhombus's diagonals to recover the full length is still Grade 5 quadrilateral reasoning.

#7 Identify Subproblems 6.G.A.1 Step 5

Apply the rhombus area formula: half the product of the two diagonals.

$$\text{Area} = \dfrac{1}{2} \times d_1 \times d_2 = \dfrac{1}{2} \times 24 \times 10 = 120 \text{ square meters} \;\Rightarrow\; \textbf{(D)}$$

💡 The half-product-of-diagonals formula for a special quadrilateral is Grade 6 area-of-polygons content.

[1] #1 3.OA.A.3 Find the side length. A rhombus has four equal sides, so dividing the perimeter

[2] #1 5.G.B.4 Sketch the rhombus with both diagonals. Because the diagonals of a rhombus are p

[3] #7 8.G.B.7 Zoom in on right triangle $\triangle AMB$. The hypotenuse is the rhombus's side

[4] #7 5.G.B.4 Double $BM$ to get the full second diagonal $BD$, since the diagonals bisect eac

[5] #7 6.G.A.1 Apply the rhombus area formula: half the product of the two diagonals.

Review

Reasonableness: Quick sanity check: the rhombus fits inside the $24 \times 10$ 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 $\sqrt{144}=12$), so $120$ is in the right magnitude. Choice (D) is confirmed.

Alternative: Tool #6 (Guess and Check) on a side-$13$ triangle: the only Pythagorean triple with hypotenuse $13$ and integer legs is $5$–$12$–$13$. Once you spot that, $BM = 5$ instantly without writing the equation. Combined with the diagonal-area formula, you reach $(1/2)(24)(10) = 120$ in just two lines.

CCSS standards used (min grade 8)

3.OA.A.3 Solve multiplication and division word problems within 100 (Dividing the perimeter $52$ by the $4$ equal sides of the rhombus to get the side length $13$.)
5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties (Using the defining properties of a rhombus — equal sides and perpendicular-bisecting diagonals — to label half-diagonals and recover full ones.)
6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing (Applying the rhombus area formula $\tfrac{1}{2} d_1 d_2$ once both diagonals are known.)
8.G.B.7 Apply the Pythagorean theorem to determine unknown side lengths in right triangles (Finding the unknown half-diagonal $BM$ from the right triangle with legs $AM = 12$ and hypotenuse $AB = 13$.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 8)

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