Question

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?

$\dfrac{25}{27}$

$\dfrac{26}{27}$

$\dfrac{73}{75}$

$\dfrac{145}{147}$

$\dfrac{74}{75}$

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.

2018 AMC 10A Problem 23

Solution:

Problem 23 in Other Years