AMC 10 · 2017 · #21

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
similar-trianglesratio-proportionpythagorean-theorem similar-trianglesconvert-to-algebra ↑ Prerequisites: similar-triangles
📏 Long solution 💡 3 insights
Problem
Take a right triangle with legs 3 and 4 and hypotenuse 5. In one copy, fit a square so a corner of the square sits at the right-angle vertex and its two near sides run along the legs; call its side x. In another copy, fit a square so one whole side lies on the hypotenuse; call its side y. Find the ratio xy\frac{x}{y}.

Pick an answer.

(A)
$\dfrac{12}{13}$
(B)
$\dfrac{35}{37}$
(C)
1
(D)
$\dfrac{37}{35}$
(E)
$\dfrac{13}{12}$

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Each square already has a name, x and y, so Tool #4 (Introduce a Variable) is the spine: in each picture a square cuts off a smaller triangle that is the same shape as the original, and that matching shape turns into a proportion you can solve for the side. Tool #7 (Identify Subproblems) splits the work cleanly — find x from the corner square, find y from the hypotenuse square, then combine — because the two triangles never touch. Tool #1 (Draw a Diagram) makes 'which little triangle is similar' obvious, which is the one thing that is easy to get wrong here.

1STEP 1

Split into two square puzzles

Solve each square on its own: each carves off a small triangle similar to the whole 3-4-5, giving one equation per square.

xy\frac{x}{y} = x1\frac{x}{1}·1y\frac{1}{y}, find x and y separately
2STEP 2

Square in the corner

The little triangle above the corner square is a shrunk 3-4-5, so its base-to-height ratio gives x3−x\frac{x}{3-x}=43\frac{4}{3}, hence x=127\frac{12}{7}.

x3−x\frac{x}{3-x}=43\frac{4}{3} → 7x=12 → x=127\frac{12}{7}
3STEP 3

Height onto the hypotenuse

Measure the same area from two bases: 12\frac{1}{2}·3·4=12\frac{1}{2}·5·h, so the height standing over the hypotenuse is h=125\frac{12}{5}.

12\frac{1}{2}·3·4=12\frac{1}{2}·5· h → h=125\frac{12}{5}
4STEP 4

Square on the hypotenuse

The square's top edge cuts a triangle similar to the whole, so yh−y\frac{y}{h-y}=5h\frac{5}{h}; with h=125\frac{12}{5} this gives y=6037\frac{60}{37}.

yh−y\frac{y}{h-y}=5h\frac{5}{h} → y=5hh+5\frac{5h}{h+5}=6037\frac{60}{37}
5STEP 5

Take the ratio

Divide the two sides: xy\frac{x}{y}=1276037\frac{\frac{12}{7}}{\frac{60}{37}}=127\frac{12}{7}·3760\frac{37}{60}=3735\frac{37}{35}, which is choice (D).

xy\frac{x}{y}=127\frac{12}{7}·3760\frac{37}{60}=3735\frac{37}{35} → (D)
Answer
3735\frac{37}{35}
The ratio 3735\frac{37}{35} is just over 1, which fits the picture: x=127\frac{12}{7}≈1.71 and y=6037\frac{60}{37}≈1.62, so the corner square is a touch bigger than the hypotenuse square. Both side lengths are smaller than the shortest leg 3, as any inscribed square must be, and both came from a similar-triangle proportion that keeps the 3-4-5 shape. The value lands exactly on choice (D).
💡Key takeaway

A square dropped into a triangle leaves a smaller copy of the same triangle, and matching their side ratios pins down the square's size.

  • Split into two square puzzles
  • Square in the corner
  • Height onto the hypotenuse
  • Square on the hypotenuse
  • Take the ratio

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