By: Rich Zwelling, Apex GMAT Instructor

Date: 7th January, 2021

## 30-60-90 Right Triangle

In a previous piece, we covered the 45-45-90 right triangle, also known as the *isosceles right triangle*. There is another so-called “special right triangle” commonly tested on the GMAT, namely the **30-60-90 right triangle**.

Like the isosceles right, its sides always fit a specific ratio, as seen in the above diagram (1 : √3 : 2). And it’s worth noting, as with all triangles, that *the shortest side is opposite the smallest angle*, while *the longest side is opposite the largest angle*, etc.

Now, it’s easy enough to memorize this ratio and deduce what each side length will be, given that we are dealing with a 30-60-90 triangle. For example, Suppose we are given the following information:

This is low-level memorization, and we can deduce that the side opposite the 60-degree angle will be length 5√3, while the hypotenuse will be length 10.

But let’s look to this GMAT Official Guide problem to see something a little more high-level. Give it a shot before reading further:

In the figure above, V represents an observation point at one end of a pool. From V, an object that is actually located on the bottom of the pool at point R appears to be at point S. If VR = 10 feet, what is the distance RS, in feet, between the actual position and the perceived position of the object?

(A) 10−5√3

(B) 10−5√2

(C) 2

(D) 2 1/2

(E) 4

(For starters, notice that the question they’re asking for — the distance between the actual position and the perceived position — is just line segment RS. Remember that the GMAT is very good at using complicated wording to frame a simple concept. Always simplify the question as quickly as possible.)

To understand this problem, let’s first talk about one of the higher-level ways the GMAT could test 30-60-90 triangles. Take this example:

Notice we are given no angles except the right angle. But we do have 2 sides and 1 angle in total, which is sufficient to form a unique triangle. Furthermore, did you identify anything that gives this away as a 30-60-90?

The hypotenuse is twice the length of one of the sides, giving them a 2:1 ratio. That guarantees that the third side fits the √3 component of our ratio, giving that side a length of 5√3. So even without labeled angles:

*A right triangle with a hypotenuse twice the length of one of its legs **must** be a 30-60-90 triangle***.**

That’s much more the kind of critical thinking the GMAT is interested in testing.

Similarly, in this Official Guide problem, we are told that VR is length 10:

Notice that at this point, it’s up to you to make the deduction that we have a 30-60-90 triangle, and thus the distance from the right angle marker to point R must be 5√3:

From there, it’s straightforward to see that RS is simply the marked length of 10 minus the length of 5√3 we just deduced, thus leading us to answer choice A.

In terms of strategy, another point: a brief look at the answer choices at the start of the problem gives a strong hint that either a 30-60-90 or 45-45-90 triangle is involved. Notice that the first two answers feature a √3 and a √2 term, and this is clearly a geometry question. This gives you the opportunity to be preemptive and use the test’s patterns against itself.

In our next post, we’ll talk about how 30-60-90 triangles can be used directly to calculate the area of equilateral triangles. You can also link to our other article about triangles:

A Short Meditation on Triangles

The 30-60-90 Right Triangle

The 45-45-90 Right Triangle

The Area of an Equilateral Triangle

Triangles with Other Shapes

Isosceles Triangles and Data Sufficiency

Similar Triangles

3-4-5 Right Triangle

5-12-13 and 7-24-25 Right Triangles

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