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 (1/2,1/2) and radius √2/2.

(x-1/2)²+(y-1/2)²=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=1/2 d₁ d₂=1/2·2·2=2
5STEP 5

Add the four semicircle caps

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

2+4·1/2π(√2/2)²=2+π=π+2=(B)
Answer
π+2
Numerically π+2≈5.14. Each quadrant circle has area π r²=π·1/2=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

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