Question

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?

$\tfrac{25}{27}$

$\tfrac{26}{27}$

$\tfrac{73}{75}$

$\tfrac{145}{147}$

$\tfrac{74}{75}$

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 $S$ is $[0, s] \times [0, s].$ The hypotenuse is $3x + 4y - 12 = 0,$ and the distance from its nearest corner $(s, s)$ is $\frac{|3s + 4s - 12|}{\sqrt{3^2 + 4^2}} = \frac{|7s - 12|}{5} = 2.$ This gives $s = \tfrac{22}{7}$ or $s = \tfrac27;$ 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 $1 - \frac{4/49}{6} = 1 - \frac{2}{147} = \frac{145}{147}.$

Thus, the correct answer is D.

2018 AMC 12A Problem 17

Solution:

Problem 17 in Other Years