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: 6th January, 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

Read more
3-4-5 triangles on the gmat
Posted on
15
Dec 2020

The 3-4-5 Right Triangle – GMAT Geometry Guide

By: Rich Zwelling, Apex GMAT Instructor
Date: 17th December, 2020

Right Triangle Identities: 3-4-5

Right triangles always adhere to the same basic relationship, reflected by the Pythagorean Theorem, or + b² = c², where a, b, and c match the triangle sides as pictured above. c always represents the longest side, called the hypotenuse.

But rather than use the formula directly, the most common way the GMAT will test knowledge of the formula is through the simplest integer values that fit this relationship. The most common is + 4² = 5² → 9 + 16 = 25, as pictured below:

What’s important to remember is that this relationship works not only for 3-4-5, but also for any corresponding multiples, such as 6-8-10 or 9-12-15 or any other multiples of the original numbers.

6-8-10 triangle on the gmat

GMAT Triangle Problem #1

If you rely solely on the formula, you could certainly get the job done, but it will take you a lot longer. Here’s an Official Guide problem that drives this point home:

The figure above shows a path around a triangular piece of land. Mary walked the distance of 8 miles from P to Q and then walked the distance of 6 miles from Q to R. If Ted walked directly from P to R, by what percent did the distance that Mary walked exceed the distance that Ted walked?

(A)   30%
(B)   40%
(C)   50%
(D)   60%
(E)   80%

If you know your so-called “Pythagorean Triples” from memory (e.g. 3-4-5, 6-8-10), this problem moves along much faster. By test day, you should know within seconds that segment PR is length 10, no calculations involved. 

After that, the bulk of your time should be spent calculating the percent difference between Mary’s total distance (14) and Ted’s total distance (10). (Answer: Since Mary walked 4 more miles more than Ted’s original 10, and 4 is 40% of 10, this makes B the correct answer.)

GMAT Triangle Problem #2

Also, it’s much more likely that the GMAT will test your knowledge of completeness of information with respect to Right Triangles, especially on Data Sufficiency. Give this problem a shot before reading on:

What is the area of triangle ABC pictured above?

  1. The length of segment AB is 5
  2. The perimeter of triangle ABC is 12

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D) EACH statement ALONE is sufficient to answer the question asked.
E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

If you know your 3-4-5 Pythagorean triple really well, you may prematurely view Statement (1) as sufficient, since you may believe the hypotenuse of 5 automatically guarantees the legs of the triangle must be 3 and 4. But this assumes the legs must have integer lengths. In reality, the legs could be any non-integer lengths that satisfy + b² = 5²

This is a classic way the GMAT could throw you off your guard. And it’s another way to test that only one side of a triangle is not enough to give you complete information about the entire triangle. Statement (1) is actually INSUFFICIENT, because we do not have information about a unique triangle, and thus could not possibly know the area.

Likewise, Statement (2) is INSUFFICIENT, because there are many ways to generate a right triangle with a perimeter of 12.

When we combine the statements, however, it’s interesting to note that, as a rule, we know we have a unique triangle if we’re given both the perimeter and the hypotenuse. As such, we would be able to find the area (even though we don’t have to calculate it), and thus the answer is C. Don’t do any math!

Takeaways

The big takeaway here is that, rather than have you use the Pythagorean theorem directly, the GMAT will try to force you into false conclusions, such as believing a hypotenuse of 5 gives you all the information you need. Be on your toes! Make sure to thoroughly examine all information given to you!

 

The 3-4-5 triangle is not the only identity to review in this triangle geometry section, here are some other identities and triangle related topics to review:
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
Triangle Inequality Rule

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gmat triangle review
Posted on
15
Dec 2020

A short meditation on GMAT triangles – Read this if you struggle with triangle problems on the GMAT!

By: Rich Zwelling, Apex GMAT Instructor
Date: 15th December, 2020

GMAT Triangles

Close your eyes, imagine a triangle, then open your eyes again.

Which of the following did you picture? Something different?

Triangle problems explained

Well, every one of the above is tested on the GMAT. And you can’t stick to just one conception of a single shape. You have to be familiar with different types of triangles and all of their characteristics. 

For example, suppose you need to find the two-dimensional area within a triangle. What’s the proper way to do this for a right triangle (2), which contains a 90-degree angle? Even if I know the proper formula, would applying it prove different if I needed to use it on triangle (4), an obtuse triangle, which has one angle greater than 90 degrees?

What about the perimeter, or the sum of the lengths of the three sides of the triangle? Are there simple, straightforward ways to calculate this in the case of the equilateral triangle (1), where each side is the same length. Or the isosceles triangle (3), where two of the three sides are the same length?

What about that isosceles triangle? Did you notice that the two sides that are the same length are opposite two angles of equal measure? What are the implications of this? 

How about the fact that all the angles in any triangle add up to the same degree measure, no matter how the triangle looks?

And then there’s something as simple as the base of the triangle. Take triangle (4):

gmat triangles

As shown, the shortest side is the base, or bottom side, of the triangle. But in truth, any side of a triangle could be considered the base. What if we took the same triangle and rotated it clockwise:

gmat triangle guide articleNow suddenly, the same triangle has the longest side as its base. This could completely change how you calculate, for example, the area of the triangle.

What about the relationships among the sides of a right triangle, for example?

It turns out that the sides a, b, and c, of all right triangles conform to a special mathematical relationship, no matter their lengths. And in its most elementary forms as well. For example, did you know that if a right triangle has short sides (legs) of lengths 3 and 4, the longest side (hypotenuse) will always be 5?

gmat triangle - right triangle

The GMAT is less interested in very complex math and much more interested in variations on basic facts like this one. 

Also, what happens when triangle types are combined? What happens when a triangle is both isosceles AND right? How does this affect the relationships among the sides?

Another way the test can give triangles (and other shapes) a twist is to combine them with other shapes. You may see questions that involve, for example, a triangle within a circles (inscribed):

triangle inscribed in a circle gmat triangle problemThis presents all sorts of opportunities for testing not only the measurements of triangles and of circles individually but also the relationships among those measurements and the ways in which they might overlap

This is of course meant to be a brief overview of the basic forms GMAT triangles can take. But the above poses critical thinking questions you should examine as you move forward with your preparation. Remember: don’t just think in terms of memorizing facts! There will be a handful of formulas to know, to be sure, but once those are out of the way, it will be up to you to be on the lookout for new ways that knowledge could be applied and tested. 

 

As the next step in your discovery and practice of GMAT triangles visit these posts:

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
Triangle Inequality Rule

 

Read more