2018 AMC 10A Problem 23

Problem 23 of 25HarderGeometry

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

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

Let

x

be the side length of

S.

Then we can split the field up into the following shapes.

We can express the area of the field in two ways:

\dfrac{3 \cdot 4}{2} = x^2 + \dfrac{x(3 - x)}{2}

+ \dfrac{x(4 - x)}{2} + \dfrac{2 \cdot 5}{2}.

Simplifying yields

6 = \dfrac{7x}{2} + 5

x = \dfrac{2}{7}.

The desired fraction is

\dfrac{6 - x^2}{6} = \dfrac{6 - \frac{4}{49}}{6} = \dfrac{145}{147}.

Thus, D is the correct answer.