M07-22

Official Solution:


Is the area of triangle ABC greater than the area of triangle DEF?

(1) The numerical value of the area of ABC is less than the numerical value of the perimeter of DEF.

Look at this statement conceptually.

Consider an isosceles right triangle ABC with sides of 1, 1, and \(\sqrt{2}\), which has an area of 1/2 and a perimeter of \(2+\sqrt{2}\). Notice that the numerical value of the area (\(0.5\)) is substantially smaller than the numerical value of the perimeter (\(2+\sqrt{2}\), which is 3.something)

Now, let's imagine another triangle DEF that is similar to ABC but slightly smaller in scale. In this case, the numerical value of the area of ABC (\(0.5\)) will still be less than the numerical value of the perimeter of DEF, which will be slightly less than \(2+\sqrt{2}\), but the area of ABC will be greater than the area of DEF, since we are considering the case where DEF is smaller in scale.

Conversely, if DEF is similar to ABC but slightly larger in scale, the numerical value of the area of ABC (\(0.5\)) will still be less than the numerical value of the perimeter of DEF, which will be slightly larger than \(2+\sqrt{2}\), but the area of ABC will be smaller than the area of DEF, since we are considering the case where DEF is larger in scale.

Therefore, the first statement is not sufficient to answer the question.

(2) Angles of ABC equal to angles of DEF.

This only indicates that the triangles are similar, but provides no information about their relative scale. Hence this statement is also not sufficient.

(1)+(2) We used isosceles right triangles to test statement (1), and since all isosceles right triangles are similar, these examples also satisfy statement (2). Therefore, even taken together the statements are not sufficient to answer the question.


Answer: E