2018 AMC 12A Problem #17

Geometry Level 3

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square S 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 S to the hypotenuse is 2 units. What fraction of the field is planted?

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73 75 \dfrac{73}{75} 25 27 \dfrac{25}{27} 26 27 \dfrac{26}{27} 145 147 \dfrac{145}{147} 74 75 \dfrac{74}{75}

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1 solution

Jerry McKenzie
Feb 9, 2018

Let the right angle be the origin

The hypotenuse is equated by 3x+4y=12.

Let the closest corner of S be (s,s).

The required distance is 3 s + 4 s 12 3 2 + 4 2 = 2 s = 2 7 \frac{3s+4s-12}{\sqrt{3^2+4^2}}=-2 \Rightarrow s=\frac{2}{7}

note we use -2 as the distance because the point is under the line.

It then follows the area of the triangle is 6, the area of S is 4 49 \frac{4}{49} .

Thus the required fraction is 6 4 49 6 = 145 147 \frac{6-\frac{4}{49}}{6}=\frac{145}{147}

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