AMC 10 · 2018 · #23

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

similar-trianglesarea-trianglespythagorean-theoremarea-difference identify-subproblemsconvert-to-algebra ↑ Prerequisites: similar-trianglesarea-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A field is a right triangle with legs 3 and 4. A small unplanted square sits in the right-angle corner, with two of its sides flush against the two legs. The square's far corner is exactly 2 units from the hypotenuse. The rest of the field is planted. Find what fraction of the field is planted.

Pick an answer.

(A)

$\frac{25}{27}$

(B)

$\frac{26}{27}$

(C)

$\frac{73}{75}$

(D)

$\frac{145}{147}$

(E)

$\frac{74}{75}$

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The square's side is unknown, so name it with a variable. The key move is to cut the triangle into pieces whose areas we can each write down — the square, two corner triangles, and one triangle resting on the hypotenuse — then force those pieces to add up to the whole. That turns the geometry into one clean equation. Finally, instead of measuring the planted region directly, find it as the whole minus the unplanted square.

1STEP 1

Name the square's side

Let s be the unplanted square's side; since two sides lie on the legs, its far corner sits s from each leg, and s controls everything.

s = side length of the square

2STEP 2

Size up the triangle

Legs 3 and 4 give hypotenuse 5 by the Pythagorean theorem, and the field's area is half of 3 times 4, namely 6.

hypotenuse = √(3² + 4²) = 5, area = 1/2· 3 · 4 = 6

3STEP 3

Triangle on the hypotenuse

Join the far corner to the hypotenuse's two ends: base 5, height 2 (the distance to the hypotenuse), so this triangle's area is 5.

[ hypotenuse triangle ] = 1/2· 5 · 2 = 5

4STEP 4

The two corner triangles

The far corner also cuts two thin triangles along the legs, with areas s(4 − s)/2 and s(3 − s)/2.

1/2s(4 - s) + 1/2s(3 - s)

5STEP 5

Make the pieces add to 6

Setting square + two corner triangles + hypotenuse triangle = 6 cancels the s² terms and leaves 7s = 2, so s = 2/7 and s² = 4/49.

s² + 1/2s(4-s) + 1/2s(3-s) + 5 = 6 → 7s/2 = 1 → s = 2/7, s² = 4/49

6STEP 6

Planted fraction

Planted is the whole minus the square, so the fraction is 1 − (4/49)/6 = 1 − 2/147 = 145/147, choice (D).

1 - (4/49)/6 = 1 - 4/294 = 1 - 2/147 = 145/147

Answer

145/147

The square is tiny: its side 2/7 is well under 1, and its area 4/49 is about 0.08 against a field of area 6. So almost all the field is planted, and 145/147 is just a hair below 1 — exactly the kind of answer we expect. As a quick filter, the unplanted share is 2/147, which forces a denominator with a factor of 147; among the choices only 145/147 has it.

💡Key takeaway

Cut the shape into pieces you can measure, make them add up to the whole, and the unknown falls out.

Name the square's side
Size up the triangle
Triangle on the hypotenuse
The two corner triangles
Make the pieces add to 6
Planted fraction