AMC 10 · 2016 · #19

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
coordinate-geometrysimilar-trianglesratio-proportion convert-to-algebra ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 3 insights
Problem
In rectangle ABCD with AB=6 and BC=3, side BC is split into three equal pieces by points E and F, so BE=EF=FC. Segment AE crosses diagonal BD at P, and segment AF crosses diagonal BD at Q. Writing BP:PQ:QD as r:s:t in lowest terms (greatest common factor 1), find r+s+t.

Pick an answer.

(A)
7
(B)
9
(C)
12
(D)
15
(E)
20

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

How to solve
Strategy Introduce a Variable

The corners are perpendicular, so tool #4 (Introduce a Variable) lets me drop the rectangle onto a coordinate grid and turn every line into an equation. Putting corner A at the origin makes AE and AF lines through the origin, which are easy to write. Tool #1 (Draw a Diagram) fixes the order B,P,Q,D along the diagonal so I know which piece is which. Tool #7 (Identify Subproblems) splits the work into two clean intersection problems — find P from AE∩ BD, then Q from AF∩ BD — and a single shared variable that runs along BD turns both crossings directly into the three ratio pieces.

1STEP 1

Place the rectangle on a grid

Put A=(0,0), B=(6,0), C=(6,3), D=(0,3); trisecting BC gives E=(6,1) and F=(6,2).

A=(0,0), B=(6,0), C=(6,3), D=(0,3), E=(6,1), F=(6,2)
2STEP 2

Write the three lines as equations

AE and AF pass through the origin: AE is y=x/6, AF is y=x/3; write BD as (6-6u, 3u), u=0 at B and u=1 at D.

AE: y=x/6, AF: y=x/3, BD: (x,y)=(6-6u, 3u)
3STEP 3

Find P, then Q, on the diagonal

Substitute (6-6u,3u) into each line: P gives 3u=1-u so u=1/4; Q gives 3u=2-2u so u=2/5.

P: 3u=1-u→ u=1/4; Q: 3u=2-2u→ u=2/5
4STEP 4

Turn the fractions into the ratio

Distances from B track u, so BP:PQ:QD=1/4:3/20:3/5=5:3:12, gcd 1, giving r+s+t=20 — choice (E).

BP:PQ:QD=1/4:3/20:3/5=5:3:12, 5+3+12=20 → (E)
Answer
20
The three parts must rebuild the whole diagonal: 5+3+12=20 matches the full u-span 0 to 1 scaled by 20, so nothing is lost or double-counted. The order also makes sense — E is closer to B than F is, so line AE should cross BD nearer B, and indeed P (u=1/4) comes before Q (u=2/5). The reduced parts 5,3,12 share no common factor, so the ratio is already in lowest terms.
💡Key takeaway

Drop the rectangle on a grid, write each line as an equation, and let one fraction u slide along the diagonal — the gaps in u are exactly the pieces 5:3:12, which add to 20.

  • Place the rectangle on a grid
  • Write the three lines as equations
  • Find P, then Q, on the diagonal
  • Turn the fractions into the ratio

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