AMC 10 · 2004 · #20

Grade 7 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
ratio-proportionarea-triangles convert-to-algebraidentify-subproblems ↑ Prerequisites: ratio-proportion
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
In triangle ABC, point D is on side BC and point E is on side AC. The cevians AD and BE cross at T. Along AD, the piece AT is 3 times the piece DT; along BE, the piece BT is 4 times the piece ET. Find how the point D splits BC, that is, the ratio CD/BD.

Pick an answer.

(A)
$\frac{1}{8}$
(B)
$\frac{2}{9}$
(C)
$\frac{3}{10}$
(D)
$\frac{4}{11}$
(E)
$\frac{5}{12}$

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The picture has no measurable lengths — only two ratios along the two cevians — so I trade lengths for areas, which the ratios control directly. The key fact (Tool #7, Identify Subproblems) is that two triangles sharing the same height have areas in the ratio of their bases. So I cut triangle ABC into the four small triangles meeting at T and let one of them be my unit. Then I name the one unknown area with a variable (Tool #4, Introduce a Variable): the area of triangle TCD. Reading triangle ACT's area two different ways gives a single equation in that variable, and that variable turns out to equal the answer CD/BD itself. Tool #1 (Draw a Diagram) keeps the four pieces and which cevian controls which ratio straight.

1STEP 1

Turn the length ratio into an area ratio

Triangles TCD and TBD share apex T and sit on line BC, so their heights match: CD/BD=[TCD]/[TBD], where [ · ] means area.

CD/BD=[TCD]/[TBD]
2STEP 2

Use the AD ratio: set a unit and a variable

Let [TBD]=1 and [TCD]=x, which is the answer. From AT:TD=3:1, apex B gives [ABT]=3 and apex C gives [ACT]=3x.

[TBD]=1, [TCD]=x; [ABT]=3, [ACT]=3x
3STEP 3

Use the BE ratio on the same pieces

Now BT:TE=4:1 gives [AET]=[ABT]/4=3/4, and since D splits CBT into 1 and x, [CET]=(1+x)/4.

[AET]=1/4 · 3=3/4, [CET]=1/4(1+x)=(1+x)/4
4STEP 4

Read triangle ACT two ways and solve

E on AC splits ACT, so 3x=3/4+(1+x)/4, hence 12x=4+x, 11x=4, and CD/BD=x=4/11, choice (D).

3x=3/4+(1+x)/4 → 12x=4+x → 11x=4 → x=4/11 (D)
Answer
4/11
The answer is a ratio between 0 and 1, which fits: D sits partway along BC, and the strong pull of the cevians (AT three times TD, BT four times ET) should keep D closer to B than to C, so CD ＞ BD would be wrong — indeed 4/11 ＜ 1 means CD ＜ BD, consistent with D near B. All areas came out positive ([TBD]=1,[TCD]=4/11,[ABT]=3,[AET]=3/4), so no piece was forced negative, and 4/11 is exactly one of the offered choices.
💡Key takeaway

Trade lengths for areas: triangles with the same height compare by their bases, so name one small triangle's area x, write another triangle's area two different ways, and the single equation hands you CD/BD=4/11.

  • Turn the length ratio into an area ratio
  • Use the AD ratio: set a unit and a variable
  • Use the BE ratio on the same pieces
  • Read triangle ACT two ways and solve

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