2018 AMC 12A Problem 17

Problem 17 of 25IntermediateGeometry

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle’s legs have lengths of

3

and

4

units. In the corner where those sides meet at a right angle, he leaves a small unplanted square

S

so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from

S

to the hypotenuse is

2

units. What fraction of the field is planted?

Show solution

Solution

Place the right angle at the origin with legs on the axes, so the vertices are

(4, 0),

(0, 3),

(0, 0),

and the square

S

[0, s] \times [0, s].

The hypotenuse is

3x + 4y - 12 = 0,

and the distance from its nearest corner

(s, s)

\frac{|3s + 4s - 12|}{\sqrt{3^2 + 4^2}} = \frac{|7s - 12|}{5} = 2.

This gives

s = \tfrac{22}{7}

s = \tfrac27;

only

s = \tfrac27

keeps the square inside the triangle. The field has area

\tfrac12 \cdot 3 \cdot 4 = 6

and the unplanted square has area

\left(\tfrac27\right)^2 = \tfrac{4}{49}.

The planted fraction is

1 - \frac{\frac{4}{49}}{6} = 1 - \frac{2}{147} = \frac{145}{147}.

Thus, the correct answer is D.

Tagged: coordinate geometry · distance formula · area