triangle inequality rule
Posted on
09
Feb 2021

Triangle Inequality Rule

By: Rich Zwelling (Apex GMAT Instructor)
Date: 9 Feb 2021

One of the less-common but still need-to-know rules tested on the GMAT is the “triangle inequality” rule, which allows you to draw conclusions about the length of the third side of a triangle given information about the lengths of the other two sides.

Often times, this rule is presented in two parts, but I find it is easiest to condense it into one, simple part that concerns a sum and a difference. Here’s what I mean, and we’ll use a SCENARIO:

Suppose we have a triangle that has two sides of length 3 and 5:

triangles inequalities 1

What can we say about the length of the third side? Of course, we can’t nail down a single definitive value for that length, but we can actually put a limit on its range. That range is simply the difference and the sum of the lengths of the other two sides, non-inclusive.

So, in this case, since the difference between the lengths of the other two sides is 2, and their sum is 8, we can say for sure that the third side of this triangle must have a length between 2 and 8, non-inclusive. [Algebraically, this reads as (5-3) < x < (5+3) OR 2 < x < 8.]

If you’d like to see that put into words:

**The length of any side of a triangle must be shorter than the sum of the other two side lengths and longer than the difference of the other two side lengths.**

It’s important to note that this works for any triangle. But why did we say non-inclusive? Well, let’s look at what would happen if we included the 8 in the above example. Imagine a “triangle” with lengths 3, 5, and 8. Can you see the problem? (Think about it before reading the next paragraph.)

Imagine a twig of length 3 inches and another of length 5 inches. How would you form a geometric figure of length 8 inches? You’d simply join the two twigs in a straight line to form a longer, single twig of 8 inches. It would be impossible to form a triangle with a side of 8 inches with the original two twigs.

triangle inequalities 2

 

If you wanted to form a triangle with the twigs of 3 and 5, you’d have to “break” the longer twig of 8 inches and bend the two twigs at an angle for an opportunity to have a third side, guaranteed to be shorter than 8 inches:

triangle inequalities 3

The same logic would hold for the other end of the range (we couldn’t have a triangle of 3, 5, and 2, as the only way to form a length of 5 from lengths of 2 and 3 would be to form a longer line segment of 5.)

Now that we’ve covered the basics, let’s dive into a few problems, starting with this Official Guide problem:

If k is an integer and 2 < k < 7, for how many different values of k is there a triangle with sides of lengths 2, 7, and k?
(A) one
(B) two
(C) three
(D) four
(E) five

Strategy: Eliminate Answers

As usual with the GMAT, it’s one thing to know the rule, but it’s another when you’re presented with a carefully worded question that tests your ability to pay close attention to detail. First, we are told that two of the lengths of the triangle are 2 and 7. What does that mean for the third side, given the triangle inequality rule? We know the third side must have a length between 5 (the difference between the two sides) and 9 (the sum of the two sides).

Here, you can actually use the answer choices to your advantage, at least to eliminate some answers. Notice that k is specified as an integer. How many integers do we know now are possible? Well, if k must be between 5 and 9 (and remember, it’s non-inclusive), the only options possibly available to us are 6, 7, and 8. That means a maximum of three possible values of k, thus eliminating answers D and E.

Since the GMAT is a time-intensive test, you might have to end up guessing now and then, so if you can strategically eliminate answers, it increases your chances of guessing correctly.

Now for this problem, there’s another condition given, namely that 2 < k < 7. We already determined that k must be 6, 7, or 8. However, of those numbers, only 6 fits in the given range 2 < k < 7. This means that 6 is the only legal value that fits for k. The correct answer is A.

Note:

It’s important to emphasize that the eliminate answers strategy is not a mandate. We’re simply presenting it as an option that works here because it is useful on many GMAT problems and should be explored and practiced as often as possible.

Check out the following links for our other articles on triangles and their properties:

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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45-45-90 triangles on the gmat
Posted on
06
Jan 2021

45-45-90 Right Triangle – GMAT Geometry Guide

By: Rich Zwelling (Apex GMAT Instructor)
Date: Jan 6 2021

45-45-90 Right Triangle

Another of the commonly tested triangles on the GMAT is the 45-45-90, also known as the isosceles right triangle. Know that term, as it could appear by name in a question.

As shown in the above diagram, the side lengths of this triangle always fit the same ratio (1 : 1 : √2) , where the legs are the same length and the hypotenuse length is √2 times the leg length. For example, if the leg lengths were 3 instead of 1, then the hypotenuse would be 3√2 instead of simply √2.

But likewise, don’t forget that you can go backwards and divide the hypotenuse length by √2 to get to the leg length. It may seem obvious, but it presents an important point: what’s more important than simply memorizing the ratio is understanding the mathematical relationship between the side lengths. This will help you avoid trouble if the GMAT happens to give you a problem that doesn’t conform to expectations.

For example, the following problem fits expectations quite nicely:

A yard in the shape of an isosceles right triangle has a hypotenuse of length 10√2. What is the area of this yard?

From this information, it’s easy enough to deduce that the leg length is 10, and we can draw a diagram that looks roughly like this:


From there, we can easily calculate the area, which is base*height / 2, or in this case 10*10/2 = 50.

But what happens if we give the problem a little twist:

A yard in the shape of an isosceles right triangle has a hypotenuse of length 10. What is the area of this yard?

Did you catch the twist? We’re used to the hypotenuse including a √2. This is what the GMAT will do. They’ll throw you off-center, and you’ll have to adjust. But this is also why we said earlier that what matters more than memorizing the ratio of sides is understanding the relationships between the sides of an isosceles right triangle…

Remember we said that, just as we multiply the leg length by √2 to get to the hypotenuse length, so we must divide the hypotenuse length by √2 to get to the leg length. That must mean each leg has length 10/√2. 

You can then take 10/√2 and multiply it by √2/√2 to de-radicalize the denominator and get (10√2) / 2, or a leg length of 5√2:

Notice again that we have a more unfamiliar form, with the √2 terms in the legs and an integer in the hypotenuse. We can’t count on the GMAT to give us what we’re used to. 

Now we can calculate the area:

Area = (base*height)/2 = (5√2)(5√2)/2 = (5*5)(√2*√2)/2 = (25)*(2) / 2 = 25

 

Problem #1

Now, to try this on your own, take a look at this Official Guide problem:

If a square mirror has a 20-inch diagonal, what is the approximate perimeter of the mirror, in inches?

(A)   40
(B)   60
(C)   80
(D)   100
(E)   120

Explanation:

This is a nice change-up, because it involves another shape. Did you notice that splitting a square along its diagonal creates two isosceles right triangles

Once you realize this, you can divide 20 by √2 to get 20/√2, then multiply top and bottom by √2 to get x=10√2.

Since the question asks for perimeter, we can multiply this by four to get 40√2. 

The final step is to realize that √2 is approximately 1.4. If we multiply 40 by 1.4, the only answer choice that possibly makes sense is 60, and thus the correct answer is B

 

After reviewing the 45-45-90 triangle identity, these further articles in the triangle geometry series will take you through more identities, each of the specific triangles and how the GMAT uses them to test your critical and creative solving skills:
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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gmat vrs gre competitive jobs on the market
Posted on
01
Sep 2020

GMAT vs GRE

GMAT vs GRE – the most competitive jobs on the market

by Apex GMAT

July 30, 2020

If you’re on the verge of pursuing a professional career in business administration, finance, marketing, marketing management, accounting, or law, etc. then taking the GMAT or GRE will be a detour along the way to the top. The type of exam you choose matters. There’s a positive correlation between test scores and future earnings; with higher test scores, you may qualify for a more competitive program, and ultimately, a more lucrative career. This article describes applications of the GMAT and the GRE in today’s labor market, as well as their similarities and differences, to help you determine which test is right for you. 

GMAT vs GRE | admissions differences

By far the most important factor to consider is which exam your desired institution accepts and prefers. 

Traditionally, the GMAT is the more common option when it comes to pursuing an MBA or a similar program at a business school. The test is specifically designed to evaluate skills that help MBA admission committees determine not only industry knowledge but also critical traits like risk and time management, problem solving under pressure, and adaptability, all of which are essential for a successful business career.

The GRE’s most distinguishing feature is its suitability for a wider variety of graduate school programs in fields such as business, education, engineering, humanities and arts, life sciences, physical sciences, and social sciences. If you’re targeting a non-MBA graduate discipline, pursuing a dual-degree, or you’re still unsure, then taking the GRE may allow you to kill two birds with one stone. It’s also worth noting that about 90% of MBA programs also accept GRE scores. 

To determine which exam will make you the most competitive, ask the institution’s admissions counselors if they prefer the GMAT over the GRE. Despite many business schools’ claims that they don’t have a preference, around 90% of applicants decide to apply with a GMAT score. This discrepancy might be the result of test takers’ desire to show admissions committees that they have a clear understanding of their graduate program goals and career aspirations. If you aren’t sure which type of graduate program you’re interested in, then the GRE might be the better option. However, if you want to make sure you will be as competitive as possible for an MBA program, then pick the GMAT. In both cases, ranking among the top performers requires rigorous test preparation.

GMAT vs GRE | structure, timing, scoring, costs

 

gmat vs gre

GMAT vs GRE | job prospects

Also consider each exam’s structure to determine which you’re more likely to perform well on.  GMAT prep will involve more focus on the quantitative section, which is more challenging than the GRE’s. MBA committees agree that an applicant’s performance on the quantitative section is one of the strongest indicators for a successful career. Conversely, the GRE’s sentence equivalence and text completion sections require a skilled command of highly sophisticated vocabulary, which may be particularly challenging to non-native English speakers.

The choice between the GMAT and the GRE may affect long term career earnings beginning at the graduate level. Applicants with strong GMAT scores are more likely to receive MBA scholarships, which are usually not available for GRE applicants. Some companies even finance GMAT tutoring and exam fees for their employees or interns as an investment that will yield long term results. When it comes down to actual labour market opportunities, however, the GMAT has an even stronger influence. Many firms, especially in consulting and finance, explicitly require a high GMAT score upon recruitment. 

Lifetime Earnings Difference

Moreover, there is a high correlation between GMAT score and post-MBA salary. Over the course of 12 years working with applicants to the top 10 MBA programs, we at Apex have been able to track their progress from pre-GMAT to their post MBA careers. With data gathered from admission consultants who work with elite programs, as well as financial data from clients who have completed their MBAs, we conducted an internal analysis of the relationship between the exam score and post MBA financial gains. After correcting for other factors, our study suggested that each ten point increment in one’s GMAT score equates to $80,000 – $90,000 (NPV) of extra lifetime earnings.

An investment in GMAT preparation can result in a successful high-paying professional career in the most competitive fields that draw MBA graduates:

  • Finance – Financial Analysts, Financial Advisors, Investment Bankers, Investment Fund Executives
  • Management – Marketing Managers, Business Operation Managers, IS Managers
  • Business Consulting – Management Analysts, Marketing Managers, Business Operations Consultants, Information Technology Directors, Operation Research Analysts and all C-level positions

GMAT vs GRE | working as a GMAT consultant

If you excel at test-taking and exam preparation, your GMAT journey can also lead you to securing a job as a GMAT instructor. The concept of private, one-on-one GMAT prep that Apex’s GMAT tutors offer is built around a customized GMAT curriculum. The goal of this approach is to work with both native and non-native English speakers to build cognitive skills that can be applied in and adapted to diverse working environments, resulting in career success.

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Posted on
14
May 2020

The Effects of Coffee on GMAT Performance

Let’s talk about caffeine and the effect of coffee on GMAT performance. Caffeine is a neuro-stimulant. Drinking coffee or tea while you prep and particularly being appropriately caffeinated when your test is a decided advantage. Caffeine is a nootropic, which means it helps you be smarter. It also helps your cognitive abilities become enhanced due to increased blood flow and oxygen flow to the brain.

Find Your Right Amount

It’s important to understand how much caffeine helps, not just to wake you up in the morning. More than that, it’s about how much caffeine is needed to get you to that a really nice steady state of alert focus-ness (where you’re making up words like alert focus-ness) where you kind of feel on top of the world and you have that gentle energy.

You want to understand exactly how much caffeine your body can take because there’s nothing worse than being over caffeinated, jittery and anxiety ridden on the exam. But if you calibrate it properly caffeine is an important part of your GMAT diet.

If you enjoyed the Effects of coffee on GMAT Prep, watch: Why a 4.0 does not equal GMAT success. 

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