AMC 10 · 2012 · #15

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area
coordinate-geometryarea-trianglessystems-of-equations convert-to-algebra ↑ Prerequisites: coordinate-geometryarea-triangles
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
Three unit squares are arranged in an L-shape: two side by side on top, and one directly below the left-hand one. A is the top-left corner of the figure, and B is the corner one unit to its right along the top edge. One segment runs from A to the bottom-right corner of the top row, and another runs from B to the bottom-left corner of the whole figure. The two segments cross at point C. Find the area of △ ABC.

Pick an answer.

(A)
$\frac16$
(B)
$\frac15$
(C)
$\frac29$
(D)
$\frac13$
(E)
$\frac{\sqrt{2}}{4}$

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Tool #1 (Draw a Diagram): the picture is a grid, so I lay a coordinate system on it and read every corner off as a point. Tool #4 (Introduce a Variable): with coordinates in hand, each segment becomes a line equation y=mx+b, and the crossing point C is found by treating its coordinates as unknowns. Tool #7 (Identify Subproblems): I split the work into two clean pieces — first locate C, then use AB as a base to get the area.

1STEP 1

Put the figure on a grid

Drop the figure onto a grid: A at the origin, right as +x, down as -y. Then A=(0,0), B=(1,0), and the segment ends are (2,-1) and (0,-2).

A=(0,0), B=(1,0), (2,-1), (0,-2)
2STEP 2

Write each segment as a line

From A to (2,-1) the slope is -1/2 through the origin, so y=-1/2x; from B to (0,-2) the slope is 2 through B, so y=2x-2.

y=-1/2x and y=2x-2
3STEP 3

Find where they cross

Set -1/2x=2x-2; doubling both sides gives -x=4x-4, so x=4/5 and y=-2/5, that is C=(4/5,-2/5).

-1/2x=2x-2 → x=4/5, y=-2/5 → C=(4/5,-2/5)
4STEP 4

Turn the corner into area

Base AB sits flat on y=0 with length 1, and the height is |-2/5|=2/5, so the area is 1/2·1·2/5=1/5 — choice (B).

Area=1/2 · 1·2/5=1/5 → (B)
Answer
1/5
The whole L-shape covers 3 square units, and △ ABC is a thin sliver tucked near the top, so an area far below 1 is expected — 1/5 fits. A quick shoelace check with A=(0,0), B=(1,0), C=(4/5,-2/5) gives 1/2|0(0-(-2/5))+1((-2/5)-0)+4/5(0-0)|=1/2·2/5=1/5, matching.
💡Key takeaway

Drop the picture onto a grid, turn each slanted segment into a line equation, solve the two lines together to find where they cross, then use the flat side AB as a base — the area comes out to 1/5, choice (B).

  • Put the figure on a grid
  • Write each segment as a line
  • Find where they cross
  • Turn the corner into area

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