Maths Olympiad Prep

Library / /278 of 520

Geometry Difficulty 3.1 AMC 10/12 Find the answer

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 33 and 44 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square SS so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from SS to the hypotenuse is 22 units. What fraction of the field is planted?

Pick one

Solution

Note that the hypotenuse of the field is 5,5, and the area of the field is 6.6. Let xx be the side-length of square S.S.
We partition the field into a red triangle, a yellow triangle, and a green triangle, as shown below:

Let the brackets denote areas. By area addition, we set up an equation for x:x:
[Red Triangle]+[Yellow Triangle]+[Green Triangle]=[Field]3x2+4x2+5⋅22=6,\begin{align*} [\text{Red Triangle}]+[\text{Yellow Triangle}]+[\text{Green Triangle}]&=[\text{Field}] \\ \frac{3x}{2}+\frac{4x}{2}+\frac{5\cdot2}{2}&=6, \end{align*}
from which x=27.x=\frac27. Therefore, the answer is [Field]−[S][Field]=6−x26=(D) 145147.\frac{[\text{Field}]-[S]}{[\text{Field}]}=\frac{6-x^2}{6}=\boxed{\textbf{(D) } \frac{145}{147}}.
~MRENTHUSIASM

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.