When faced with Geometry problems with variables, many test takers will approach the question with fear, believing they are forgetting some obscure geometric rule that is the only path towards a correct answer.
In reality, as we’ve covered in a few past posts, the understanding required to do well on Geometry questions on the GMAT is basic – Pythagorean theorem, special right triangles, area formula, and the like that you’ve found in the first half of an introductory Geometry course. Occasionally, we see some oddball questions associated with central angles, but there are still multiple ways to get to the correct answer.
But beyond these simple rules, the majority of Geometry problems in the GMAT Quantitative section are Algebra questions in disguise, requiring you understand what the simple geometry concept is that is being tested, then using Algebra-related problem solving from there.
Let’s examine a question that fits this mold:
In an xy-coordinate plane, a line is defined by y = kx + 1. If (4, b), (a, 4), and (a, b +1) are three points on the line, where a and b are unknown, then k = ?
- ½
- 1
- 1 and ½
- 2
- 2 and ½
Initially, we may think that this problem is impossible – if a and b are unknown, how in the world will we find a third variable? But, if we assess the answer choices, we’ll notice that these are concrete values, and the two value questions suggest that we should consider an algebraic route, perhaps with factoring?
Either way, the best way to tackle these questions is to start with what you know, then work it out from there – you’ll never be successful by trying to jump to the right answer choice.
So, what do we know here? We know that the line is y = kx + 1 and we have three points we can plug in, even if they do consist of variables. We will start with (4,b):
y = kx + 1
b = k(4) + 1
Then (a,4):
y = kx + 1
4 = k(a) + 1
Then (a, b+1):
y = kx + 1
b + 1 = k(a) + 1
That gives us three equations:
b = 4k + 1
4 = k(a) + 1
b + 1 = k(a) + 1
The trigger here is to remember the answer choices are actual values, so we need to figure out how to get the variables to cancel out. It doesn’t look like there are any squared values, so at the very least, we should be able to eliminate (C) and (E) for answer choices.
One route to take is to solve for k(a):
k(a) = 3
Then, b + 1 = 3 + 1 , making b = 3
From there,
3 = 4k + 1
2 = 4k
2/4 = k > ½
Therefore the correct answer is (A).
Many “Geometry” GMAT problems will play out this way. We are excited to have a basic understanding of lines and coordinate points, but from there, it is all about setting up equations to find a value. Whenever tackling Geometry problems with lots of values, just remember to cancel it out!
