M23-24

Official Solution:


If the area of a rectangle is 80, what is the angle between the diagonal of the rectangle and its longer side?

Important piece of information that will help to solve this question:

In a “what is the value?” GMAT DS question, your primary goal is not to calculate the value, but to determine whether you can do so with the information provided. Therefore, obtaining a numerical answer is not always necessary for this type of question.

In a "yes/no” GMAT DS question, your primary goal is not to calculate a value, but to determine whether the information provided is sufficient to answer the question with a "yes" or "no" response. Therefore, obtaining a numerical answer is not always necessary for this type of question.

Let the length and width of the rectangle be \(a\) and \(b\), respectively. We are given that \(ab = 80\) and asked to find the angle between the diagonal of the rectangle and its longer side.

(1) The perimeter of the rectangle is 84.

This statement tells us that \(2(a + b) = 84\), or that \(a + b = 42\). We can solve the system of equations \(ab = 80\) and \(a + b = 42\) to get unique values of \(a\) and \(b\). This means that the rectangle is fixed at a unique configuration, and we can answer any question regarding that rectangle. Therefore, statement (1) is sufficient.

(2) The shorter side of the rectangle is 2.

This statement implies that the longer side is 40. Again, the rectangle is fixed at a unique configuration, and we can answer any question regarding that rectangle. Therefore, statement (2) is also sufficient.


Answer: D