AMC 10 · 2007 · #21

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
similar-trianglespythagorean-theoremarea-triangles convert-to-algebra ↑ Prerequisites: similar-triangles
📏 Medium solution 💡 3 insights
Problem
In right triangle ABC with legs AB=3 and BC=4 and hypotenuse AC=5, a square XYZW is placed so that one whole side XY lies flat on the hypotenuse AC, while the opposite two corners reach the legs: W touches AB and Z touches BC. Find the side length of that square.

Pick an answer.

(A)
$\frac{3}{2}$
(B)
$\frac{60}{37}$
(C)
$\frac{12}{7}$
(D)
$\frac{23}{13}$
(E)
2

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The square rests on the hypotenuse, so the useful measurement is not a leg but the height of the triangle measured perpendicular to that hypotenuse. That splits the work into two subproblems (Tool #7): first find the altitude from B down to AC, then see how the square eats into that altitude. The altitude comes free from area (the same triangle area written two ways). For the second part, name the side s (Tool #4): the top edge of the square cuts off a smaller triangle at the top that is a scaled copy of the whole triangle, and matching the scale factor gives one equation in s.

1STEP 1

Find the height onto the hypotenuse

Area two ways: 12⋅3⋅4=6=12⋅5⋅h\frac{1}{2}\cdot3\cdot4=6=\frac{1}{2}\cdot5\cdot h, so the altitude from B onto AC is 125\frac{12}{5}.

1/2 · 3 · 4 = 6 = 1/2 · 5 · h → h = 12/5
2STEP 2

Match the small top triangle to the whole

Let s be the side. The triangle above the square is a scaled copy of ABC: base s, height 125−s\frac{12}{5}-s. So s:125−s=5:125s:\frac{12}{5}-s=5:\frac{12}{5}.

s/(12/5-s) = 5/12/5
3STEP 3

Solve for the side length

Cross-multiply: 125s=12−5s\frac{12}{5}s=12-5s, gather the s terms to get 375s=12\frac{37}{5}s=12, so s=6037s=\frac{60}{37} — choice (B).

12/5s = 12 - 5s → 37/5s = 12 → s = 60/37
Answer
60/37
The side must be smaller than the altitude onto the hypotenuse, 125=2.4\frac{12}{5}=2.4, and 6037\frac{60}{37}≈ 1.62 is comfortably below that. A quick sanity check on the formula: for a square sitting on a base b in a triangle of height h, the side is bhb+h\frac{bh}{b+h}. Here b=5b=5 and h=125h=\frac{12}{5} give 125+125\frac{12}{5+\frac{12}{5}}, and 5+125=3755+\frac{12}{5}=\frac{37}{5}, so the value is 6037\frac{60}{37} — matching. Choice (C) 127\frac{12}{7} is what the same formula returns for the legs 3 and 4, the square on a leg rather than on the hypotenuse.
💡Key takeaway

When a square sits on the slanted side, measure the triangle's height straight out from that side, then the leftover triangle on top is a smaller copy that hands you one equation.

  • Find the height onto the hypotenuse
  • Match the small top triangle to the whole
  • Solve for the side length

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