The wording on this question is a little bit strange - we shouldn't be able to directly compare a 2-dimensional area to a 1-dimensional perimeter. But - here's the logic behind this question....
Target question: is the area of ABC > area of DEF?
(1) The value of area of ABC is less than that of perimeter of DEF.
Here, test different values to try to prove insufficiency.
Scenario 1: ABC is an isosceles right triangle with a base of 1 and a height of 1, thus an area of 1/2. DEF is an isosceles right triangle of base 10 and height 10, thus an area of 50 and a perimeter of 20 + 10(sqrt2). DEF has both the greater perimeter and the greater area, so the answer is NO.
Scenario 2: ABC is an isosceles right triangle with a base of 2 and a height of 2, thus an area of 2. DEF is an isosceles right triangle of base 1 and height 1, thus an area of 1/2 and a perimeter of 2 + (sqrt2). DEF has the greater perimeter, but ABC has the greater area, so the answer is YES.
Insufficient.
(2) Angles of ABC = Angles of DEF
If all angles are the same, the triangles are similar, but that doesn't tell us anything about scale. Insufficient.
(1) & (2) Together:
In the examples we used in statement 1, all of the triangles were 45-45-90, but that did not allow us to compare area to perimeter. Insufficient.
The answer is E.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
