AMC 10 · 2013 · #18

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
coordinate-geometryarea-triangleslinear-equations-two-var convert-to-algebra ↑ Prerequisites: coordinate-geometry
📏 Long solution 💡 3 insights
Problem
Four points A=(0,0), B=(1,2), C=(3,3), D=(4,0) form quadrilateral ABCD. A line through A splits it into two pieces of equal area. That line meets side CD at a point (p/q,r/s) in lowest terms. Find p+q+r+s.

Pick an answer.

(A)
54
(B)
58
(C)
62
(D)
70
(E)
75

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The heart of the problem is the unknown crossing point on CD, so Tool #4 (Introduce a Variable) is primary: call the point E=(x,y) and turn "equal areas" into an equation for it. Tool #1 (Draw a Diagram) is used first to see that the line from A to a point on CD cuts off triangle AED sitting on the x-axis. Tool #7 (Identify Subproblems) splits the work into two clean pieces — find the whole area, then match half of it. Tool #13 (Convert to Algebra) writes side CD as a line equation so the point's x-coordinate can be solved once its height is known.

1STEP 1

Plot the points and the cut

A and D sit on the x-axis, so AD is flat with length 4. A cut from A to a point E on CD makes triangle AED plus piece ABCE.

A=(0,0), D=(4,0)→ AD on x-axis, AD=4
2STEP 2

Find the whole area

Shoelace on A,B,C,D gives 0-3-12+0=-15, so the area is 15/2. Each equal piece must then be 15/4.

[ABCD]=1/2 |{-15}|=15/2 → half=15/4
3STEP 3

Set the triangle's area to half

Let E=(x,y). Triangle AED has base AD=4 on the axis, so its area is 2y. Set 2y=15/4 to get y=15/8.

[AED]=1/2·4 · y=2y=15/4 → y=15/8
4STEP 4

Locate the point on side CD

E lies on CD, whose line is y=-3x+12. Putting y=15/8 in gives 3x=81/8, so x=27/8 and E=(27/8,15/8).

y=-3x+12, 15/8=-3x+12 → x=27/8
5STEP 5

Add the reduced parts

Both 27/8 and 15/8 are already reduced, so p+q+r+s=27+8+15+8=58, choice (B).

p+q+r+s=27+8+15+8=58 → (B)
Answer
58
Check the height is possible: side CD falls from y=3 at C down to y=0 at D, and 15/8=1.875 lies between them, so E really is on the segment (its x=27/8=3.375 is between 3 and 4 too). Confirm equal areas: triangle AED has area 2·15/8=15/4, exactly half of 15/2, so the other piece is the matching half. The parts sum to 58, choice (B); the other choices don't come from any reduced form of this point.
💡Key takeaway

Since AD lies flat on the x-axis, the cut's triangle has area just 2 × its height, so make that equal half the total, then ride the line of CD to find the exact point.

  • Plot the points and the cut
  • Find the whole area
  • Set the triangle's area to half
  • Locate the point on side CD
  • Add the reduced parts

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