AMC 10 · 2018 · #17

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)

$\frac{74}{75}$

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

Give the square's side a letter.

s = side length of the square

2STEP 2

Size up the triangle

Find the triangle's hypotenuse and area first.

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

3STEP 3

Triangle on the hypotenuse

The distance two gives the triangle on the hypotenuse.

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

4STEP 4

The two corner triangles

The two corner triangles use the same letter.

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

5STEP 5

Make the pieces add to 6

The pieces adding to the whole fixes the side.

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

6STEP 6

Planted fraction

Subtracting gives one hundred forty-five over one hundred forty-seven.

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