smc

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? FIGURE

Accepted Answer

Note that the hypotenuse of the field is 5, and the area of the field is 6. Let x be the side-length of square S. We partition the field into a red triangle, a yellow triangle, and a green triangle, as shown below: FIGURE Let the brackets denote areas. By area addition, we set up an equation for x: align* [Red Triangle]+[Yellow Triangle]+[Green Triangle]&=[Field] 3x2+4x2+522&=6, align* from which x=27. Therefore, the answer is [Field]-[S][Field]=6-x^26=145147.