M21-15

Official Solution:


Is the perimeter of a rectangle greater than 8 inches?

The question asks whether \(Perimeter=2(a+b) \gt 8\), or whether \(a+b \gt 4\), (where \(a\) and \(b\) are the lengths of the sides of the rectangle).

(1) The diagonal of the rectangle is twice as long as its shorter side.

Clearly insufficient, we know the shape of the rectangle but not its size.

(2) The diagonal of the rectangle is 4 inches longer than its shorter side.

This statement basically says that the length of the diagonal is greater than 4 inches: \(d \gt 4\). Now, consider the triangle made by the diagonal and the two sides of the rectangle: since the length of any side of a triangle must be smaller than the sum of the other two sides, then we have that \(a+b \gt d\), so \(a+b \gt d \gt 4\). Sufficient.


Answer: B