AMC 10 · 2016 · #21

Grade 8 geometry-2d

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

absolute-valuecoordinate-geometryarea-circlessymmetry-argument completing-the-squarecasework ↑ Prerequisites: coordinate-geometry

📏 Long solution 💡 3 insights

Problem

Look at every point (x,y) that satisfies x²+y²=|x|+|y|. Together these points trace a closed curve. Find the area of the region it encloses.

Pick an answer.

(A)

$\pi+\sqrt{2}$

(B)

$\pi+2$

(C)

$\pi+2\sqrt{2}$

(D)

$2\pi+\sqrt{2}$

(E)

$2\pi+2\sqrt{2}$

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The absolute values make the curve look fearsome, but they only encode symmetry: the equation is unchanged when x→-x or y→-y, so the whole picture is four mirror-image copies of one quadrant (Tool #1). That lets us solve a single quadrant and reuse it. In the first quadrant the bars vanish and the equation is a quadratic; completing the square (Tool #4) turns it into a recognizable circle. Once we see the shape, Tool #7 (Identify Subproblems) splits the enclosed region into easy pieces — a central square plus four semicircular caps — whose areas we can add.

1STEP 1

Use the symmetry to study one quadrant

The equation sees only |x| and |y|, so the curve mirrors across both axes; solve one quadrant, where it becomes x²+y²=x+y.

x ≥ 0, y ≥ 0: x²+y²=x+y

2STEP 2

Complete the square into a circle

Complete the square: x²-x+y²-y=0 becomes a circle with center ( $\frac{1}{2}$ , $\frac{1}{2}$ ) and radius $\frac{√2}{2}$ .

(x-

\frac{1}{2}

)²+(y-

\frac{1}{2}

)²=

\frac{1}{2}

3STEP 3

Spot the diameter through the axis points

The circle hits (0,0), (1,0), (0,1); the chord (1,0)-(0,1) has length √2, equal to the diameter, so the far arc is a semicircle.

dist((1,0),(0,1))=√2 = 2r

4STEP 4

Build the central square

The four axis points form a 45°-tilted square whose diagonals both equal 2, giving area 2; the region is this square plus four caps.

square area=

\frac{1}{2}

d₁ d₂=

\frac{1}{2}

·2·2=2

5STEP 5

Add the four semicircle caps

Each edge is the diameter of a radius- $\frac{√2}{2}$ semicircle of area $\frac{π}{4}$ ; four of them total π, so the whole region has area 2+π — choice (B).

2+4·

\frac{1}{2}

π(

\frac{√2}{2}

)²=2+π=π+2=(B)

Answer

π+2

Numerically π+2≈5.14. Each quadrant circle has area π r²=π·

\frac{1}{2}

\frac{π}{2}

≈1.57, so four separate disks would total 2π≈6.28; the enclosed region must be less because the disks overlap near the origin, and 5.14 ＜ 6.28 fits. The answer also lands between the smallest choice π+√2≈4.56 and the larger 2π choices, exactly where a moderately-overlapping union belongs, and matches choice (B).

💡Key takeaway

Strip away the absolute values to see four mirrored circles, then rebuild the shape as one tilted square plus four half-circle bumps: 2+π.

Use the symmetry to study one quadrant
Complete the square into a circle
Spot the diameter through the axis points
Build the central square
Add the four semicircle caps

Use the symmetry to study one quadrant

Complete the square into a circle

Spot the diameter through the axis points

Build the central square

Add the four semicircle caps

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