GMAT Question of the Day - Problem Solving - Geometry
If s is between 0 and 9, how many different equilateral triangles with side s can be formed that have an area which is an integer value?
GMAT Question of the Day Solution
This GMAT question of the day is tough but not impossible. What makes it tough? You have to consider a few different things while keeping in mind the limitations imposed by the question. First off - what's the area of an equilateral triangle? Now consider what numbers will create an integer value for the area? We have to get rid of the 4 in the denominator and the √3 in the numerator. Let's look at one issue at a time:
1. To cancel the 4 you need to have a at least a 4 in the numerator. It doesn't need to be a 4 but could be an 8 or a 12 or a 16. Anything with at least one 4 as a factor. Keep in mind the limitation that S must be less than 9. So that leaves 2, 4, 6, and 8 each of which when squared will have at least one 4.
2. We also need to either cancel or transform the √3. We can do this by either dividing or multiplying by √3. This means that S must have 3^1/4 in the numerator or the denominator.
Putting both together you have these options for S:
2(3^1/4), 4(3^1/4), 6(3^1/4)
2/(3^1/4), 4/(3^1/4), 6/(3^1/4), 8/(3^1/4)
Note that 8(3^1/4) will not work because it is greater than 9. That leaves seven different triangles.