AMC 10 · 2009 · #17

Grade 6 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Axis-Aligned Figure Area from Coordinates

area-trianglescoordinate-geometrylinear-equations-one-var identify-subproblems ↑ Prerequisites: area-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Five unit squares are placed in the coordinate plane with the lower-left corner at the origin, forming the figure shown. A slanted line runs from the point (c,0) on the x-axis up to (3,3), and this line splits the whole figure into two pieces of equal area. Find the value of c.

Pick an answer.

(A)

$\frac12$

(B)

$\frac35$

(C)

$\frac23$

(D)

$\frac34$

(E)

$\frac45$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The shaded region has an awkward staircase edge, but it becomes easy once broken into pieces. Tool #7 (Identify Subproblems) is primary: fill in the one missing unit square at the bottom-right so the shaded part plus that square is a single clean triangle, then the shaded area is just (triangle) minus (one square). Tool #1 (Draw a Diagram) reads the exact corner coordinates off the figure so the triangle's base and height are known. Tool #4 (Introduce a Variable) keeps c as the unknown, writes the triangle's area in terms of c, and turns the equal-area condition into one linear equation to solve.

1STEP 1

Find each region's target area

Five unit squares give total area 5, so each piece — the shaded one included — must have area 2.5.

total=5, each half=5/2=2.5

2STEP 2

Complete the shaded part to a triangle

The bottom-right cell is empty; fill it in and the shaded part becomes the whole triangle (c,0), (3,0), (3,3), so shaded = triangle - 1.

shaded=area of △((c,0),(3,0),(3,3))-1

3STEP 3

Write the triangle's area with c

That right triangle has base 3-c on the x-axis and height 3, so its area is 1/2(3-c)(3).

area of △=1/2(3-c)(3)

4STEP 4

Set up the equation and solve

Shaded 2.5 forces the triangle to be 3.5, so 3(3-c)=7 and 3-c=7/3, giving c=2/3, choice (C).

1/2(3-c)(3)=3.5→ 3(3-c)=7→ 3-c=7/3→ c=2/3

Answer

2/3

Plug c=2/3 back in. The triangle base is 3-2/3=7/3 and height 3, giving area 1/2·7/3·3=7/2=3.5; subtracting the one empty square leaves shaded area 2.5, exactly half of 5. As a second check, the shaded polygon (2/3,0),(3,3),(3,1),(2,1),(2,0) has area 2.5 by the shoelace formula, agreeing. The value 2/3 is between 0 and 3, so the line really does start on the figure's bottom edge, which is sensible.

💡Key takeaway

Fill in the missing square to turn a jagged shape into a clean triangle, find its area, then subtract the square back.

Find each region's target area
Complete the shaded part to a triangle
Write the triangle's area with c
Set up the equation and solve

Find each region's target area

Complete the shaded part to a triangle

Write the triangle's area with c

Set up the equation and solve

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