AMC 10 · 2004 · #14

Grade 10 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Parallel-Cut Area Ratio
similar-trianglespythagorean-theoremarea-difference identify-subproblemscomplementary-counting ↑ Prerequisites: similar-triangles
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A triangle has sides 13, 5 and 12, and points M and N sit 4 units from the vertex where the two shorter sides meet. Perpendiculars from M and N hit the long side at J and K. Find the area of the five-sided region.

Pick an answer.

(A)
15
(B)
$\frac{81}{5}$
(C)
$\frac{205}{12}$
(D)
$\frac{240}{13}$
(E)
20
How to solve
Strategy Change Focus / Count the Complement

A pentagon has no area formula, but this one is a whole triangle with two corner triangles sliced off, so Tool #16 (Change Focus / Count the Complement) makes the target the easy shape minus two easy shapes. Tool #1 (Draw a Diagram) supplies the fact that makes everything cheap: the sides 5, 12, 13 force a right angle at C, and the two perpendiculars create two more right triangles that share an angle with the big one. Tool #7 (Identify Subproblems) then handles each corner separately: find its similarity ratio to triangle ABC, square that ratio for the area, and subtract. Before subtracting, the diagram also has to confirm the two corners really are separate pieces inside the triangle.

1STEP 1

Spot the right angle at C

The side lengths reveal a right angle, so the whole area is 30.

5²+12²=169=13² → ∠ C=90°, [ABC]=1/2 · 5 · 12=30
2STEP 2

Find the two corner triangles

The two corners cut off are similar copies of the whole triangle.

△ AMJ ∼ △ ABC with ratio AM/AB=1/13; △ BNK ∼ △ BAC with ratio BN/BA=8/13
3STEP 3

Check the corners do not overlap

Placing the feet shows the corners do not overlap, so subtracting is legal.

AJ=5/13, BK=96/13, AJ+BK=101/13 ＜ 13=AB
4STEP 4

Scale the areas by the square of the ratio

Areas scale by the square of the length ratio, giving small fractions of 30.

[AMJ]=(1/13)² · 30=30/169, [BNK]=(8/13)² · 30=1920/169
5STEP 5

Subtract the two corners

Subtracting both leaves 240/13, choice (D).

[CMJKN]=30(1-1/169-64/169)=30·104/169=30·8/13=240/13 (D)
Answer
240/13
The value 240/13≈ 18.46 sits between 0 and the triangle's area 30, as any sub-region must. It also clears an easy lower bound: the pentagon contains triangle CMN, whose legs are CM=CN=4, giving area 1/2 · 4 · 4=8, plus the whole slab between MN and AB. The two removed corners total (30+1920)/169=1950/169≈ 11.54, and 30-11.54≈ 18.46 matches. Among the choices, 15, 81/5=16.2 and 205/12≈ 17.08 are all too small once the corner areas are computed, and 20 is too large, so 240/13 is the only fit.
💡Key takeaway

The pentagon is the 5-12-13 triangle with two corner triangles sliced off, and each corner is a scaled copy of the whole triangle, so square its hypotenuse ratio to get its share of the area and subtract.

  • Spot the right angle at C
  • Find the two corner triangles
  • Check the corners do not overlap
  • Scale the areas by the square of the ratio
  • Subtract the two corners

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