In the figure above, if MNOP is a trapezoid and NOPR is a parallelogram, what is the area of triangular region MNR? GMAT Explanation

In the figure above, if MNOP is a trapezoid and NOPR is a parallelogram, what is the area of triangular region MNR?

(1) The area of NOPR is 30
(2) The area of the shaded region is 5

When I first saw this question on Exam Pack 1 I didn't think much of it. Strangely though it comes up a ton in GMAT tutoring sessions even with students who are solid on the quant. The name of the game: stay organized, make the easy inferences, and avoid any assumptions/fake rules.

What are we looking for? The area of triangular region MNR

OK - easy. You know the formula for the area of a triangle right? So start there and think about what blanks you are going need to fill in.

Anything we can add to the figure?

Yes! Both a parallelogram and a trapezoid have symmetry that we can use to our advantage. Also, MNR has two angles the same so the sides opposite those angles must also be the same.

If you drop an altitude at N you'll split MNR in half (because of the two equal angles). Each of those halves is equal to the triangles QOP. You can go further and add a line at O dropping down to line MP and you'll have two equal triangles on either side of the figure. I added all of this to the diagram below.

In the figure above, if MNOP is a trapezoid and NOPR is a parallelogram, what is the area of triangular region MNR? Diagram

(1) The area of NOPR is 30

Knowing the area of parallelogram doesn't tell us the dimensions of the parallelogram or more importantly the Base/Height of the triangle. We can make an infinite number of different parallelograms with area 30. Those variations will create different areas for the triangular regions.

You can think of this at the extremes. Let's say parallelogram NOPR had a base of 1000 and a height of .03 (area of 30). And the parallelogram only has a very slight angle to it. The triangle will have a tiny area. You can push things in the other direction as well. Give NOPS a height of 1000 and a big angle. The triangle will have a much greater area. You can see in the diagram below that depending on the dimensions and angle of the parallelogram you can make a bigger triangle with a smaller parallelogram.

In the figure above, if MNOP is a trapezoid and NOPR is a parallelogram, what is the area of triangular region MNR? Diagram

So to sum it up: just because you know the are of a parallelogram doesn't mean you know the area of the triangles in a parallelogram.

INSUFFICIENT.

(2) The area of the shaded region is 5

If we know the area of the shaded region then we can just double that to get the area of triangular region MNR. SUFFICIENT.

Correct Answer: B

More GMAT Data Sufficiency Geometry Practice Questions!

This is from the GMAT Official Guide. It's a tough one that has both an algebraic component and a "distortion"/testing extremes component which is similar to what we did for statement 1 above (to prove insufficiency). In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA?

Here are two similar-ish DS Geometry questions from GMAT Question of the day:

GMAT Question of the Day Data Sufficiency Geometry 1 

GMAT Question of the Day Data Sufficiency Geometry 2 

 

In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA? GMAT Explanation

In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA? diagram

In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA?

(1) (AC)^2 = 2(AD)^2

(2) ∆ABC is isosceles.

This is a tough GMAT Data Sufficiency question from the Official Guide. We'll be attacking it with Yes/No and distortion. But first let's make sure to read carefully and define the question. Get a sense for the diagram and make some inferences. We want to know whether the area of triangular region ABC is equal to the area of triangular region DBA. We have two right triangles so let's keep in mind that right triangles have certain properties and could be special right triangles, 45-45-90 (right isosceles) or 30-60-90 or some special ratio 3-4-5, 5-12-13.

Not much to do with the diagram so let's jump into statement (1) (AC)^2 = 2(AD)^2

We can pick some values that follow that ratio. Let's make AD = 1 so AC must be √2. Now let's go ahead and do a Yes/No. You might think, hmmm, there are a million value I could pick. This is getting confusing... That's true. However, let's keep it easy. Especially because you know that you have two right triangles. Use special triangles as the side ratios will be defined for you. Also, GMAT loves special triangles so they are often a great bet for scenarios on GMAT DS. So let's try a 45-45-90 configuration and see where that takes us and then a 30-60-90 and see if that agrees or not.

If AC = √2 then CB also equals √2 leaving the area for triangular region ABC at 1. Because of 45-45-90 (or Pythagorean theorem) AB equals 2. So, with AD = 1 and AB = 2 the area of triangular region DBA is also 1. So in this case the area of triangular region ABC  is equal to the area of triangular region DBA. That gives us a clear "Yes". But that's only one scenario. For GMAT DS we have to try to prove insufficiency. Now let's see what happens with a 30-60-90.

AC = √2 CB = √6 so the area is √12/2

AD = 1 AB = 2√2 (because of 30-60-90 rules) so the area is 2.

With a 30-60-90 example the area of triangular region ABC  is not equal to the area of triangular region DBA. Because we proved a "yes" and a "no" statement 1 is insufficient.

Let's take a look at statement (2)∆ABC is isosceles.

For this one we can use "distortion". Keep triangular region ABC the same but see if you can change the size of triangular region DBA. You'll notice that you can grossly distort DBA clearly changing the area. Because you can change the area of DBA while leaving ABC unchanged statement (2) is insufficient.

Now let's put the statements together. Statement (2) constrains us to 45-45-90 and statement (1) tells us that if the triangular region ABC is a 45-45-90 then the area of triangular region ABC is equal to the area of triangular region DBA. We have a clear "yes" and there are no other possibilities so together the statements are sufficient. The answer is C.

In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA? full solution diagram

There are two GMAT Data Sufficiency strategies we worked on here:

  1. Yes/No. With a question that asks for a yes or no answer organize your solution into Yes/No. Write Yes and No on your paper to remind yourself that you need to try create scenarios that prove a yes and a no.
  2. Distortion. This can be very helpful for getting through tough geometry questions very quickly. Try to manipulate the shape while respecting whatever constraints there are in place. If you can shift the size of one shape vs another or one angle vs another you can easily prove insufficiency without doing any calculations.

Additional GMAT Data Sufficiency Yes/No, Distortion, Geometry Practice Questions

Here are a bunch of DS Yes/No examples. A couple are very similar a couple are somewhat different but for all of them you can used either Yes/No or Distortion. I'd work through all of these to master those concepts.

Here is another challenging geometry question with right isosceles triangles: The perimeter of a certain isosceles right triangle is 16 + 16√2. What is the length of the hypotenuse of the triangle?

Here's a very similar Geometry Yes/No Distortion question from GMAT Question fo the Day

Here's another GMAT Question of the Day Geometry Distortion example.

And one more GMAT Question of the Day Yes/No example.

Here's an Official GMAT question with Yes/No strategy: In the xy-plane, region R consists of all the points 

This GMAT Question of the Day Geometry example isn't really a Yes/No question because you are looking for a specific value but you can still use Yes/No and Distortion to prove insufficiency

Also, Yes/No isn't just for Geometry. Here's a Yes/No divisibility example from the GMAT Official Guide

 

In the figure above, point O is the center of the circle and OC = AC = AB. What is the value of x?

In the figure above, point O is the center of the circle and OC = AC = AB. What is the value of x?

(A) 40
(B) 36
(C) 34
(0) 32
(E) 3

This is a GMAT geometry question from the GMAT official guide for quantitative review. We get this one a lot in GMAT tutoring sessions so went ahead and made an in depth explanation. Video solution below.

This is a good one to review because it brings up an important GMAT geometry rule: the exterior angle theorem! Eh? Sounds complicated but it's actually pretty simple. Angle ACB above is an "exterior angle" of triangle AOC. The exterior angle theorem states that the exterior angle ACB equals the sum of the "remote interior angles" OAC and AOC. Basically, the exterior angle equals the sum of the angles that it's not connected to, the other two angles of the triangle known as the "remote interior angles".

Always watch out for this rule if you're dealing with triangles that share sides because: if the triangles share sides it's not unlikely that you'll have an exterior angle and will be able to use (and will probably need to use) the exterior angle theorem.

For this question we don't have much to define. We need to find the value of x.

First step in this:

  1. Draw a diagram and make easy inferences. The first one to label is that OC = AC = AB.
  2. Once you have those equal sides you can also label angles that are equal as sides that are equal within the same triangle have angles opposite them that are also equal.
  3. Now you should be thinking, hmmm, triangles sharing sides maybe there's an exterior angle. Yes, of course there is! Angle ACB. ACB is equal to AOC + OAC.
  4. Now you should also be wondering why this triangle is in a circle. How can the circle help you make an inference? For GMAT circle questions always think about "helpful radii". There's usually a radius whether it's drawn or you have to draw it that will help. In this case OA and OB are both radii so they are equal boom!
  5. That's it. You should have all angles defined by x. Go ahead and sum all of the x's  and set that equal to 180 to solve.

In the figure above, point O is the center of the circle and OC = AC = AB. What is the value of x? Solution

In the figure above, each side of square ABCD has length 1, the length of line segment CE is 1, and the length of line segment BE is equal to the length of line segment DE. What is the area of the triangular region BCE?

(A) 1/3
(B) √2/4
(C) 1/2
(D) √2/2
(E) 3/4

This is a tough geometry question from the GMAT prep tests (I think it's from exam pack 1, tests 3/4). The first time I saw this one I made the mistake of assuming a 3 dimensional figure. The question is much easier once you put the shape in 2 dimensions dropping point E onto the diagonal of square ABCD. At that point it becomes a set of somewhat straightforward inferences. Again, this is a very difficult question and a great majority of people won't get it correct. So don't worry if you get a little lost. What's key is seeing that the solution is basic. There's isn't any difficult math involved. The tough part is figuring out what the question is telling you.

What is the area of the triangular region BCE? Let's figure it out!

There are two important inferences to make:

  1. For a square always define the diagonal. It comes up a ton on GMAT geometry questions.
  2. If you're looking for the area of a triangle you'll need a height. If not dealing with a right triangle go ahead and always draw an altitude/height.

Once you've done those two things finding the area of the triangular region BCE is a piece of cake.

In the figure above, each side of square ABCD has length 1, the length of line segment CE is 1, and the length of line segment BE is equal to the length of line segment DE.

More challenging GMAT geometry questions with right isosceles triangles

Here is another challenging geometry question with 45-45-90 triangles: The perimeter of a certain isosceles right triangle is 16 + 16√2. What is the length of the hypotenuse of the triangle?

And another: In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA?

And finally a right isosceles triangle example from GMAT question of the day.

 

In the figure above, equilateral triangle ABC is inscribed in the circle. If the length of arc ABC is 24, what is the approximate diameter of the circle?

(A) 5
(B) 8
(C) 11
(D) 15
(E) 19

It's key to know that an equilateral triangle has equal sides and equal angles measuring 60 degrees each. So each angle of an equilateral triangle inscribed in a circle cuts off 1/3 of the circle. Why is that? Review the basics! An angle on the circumference of a circle cuts off twice that angle's measure. So a 60 degree angle on the edge of a circle cuts off 120 degrees or 1/3 of a circle (120/360 degrees). A 60 degree angle in the center of a circle only cuts off 60 degrees (it doesn't have the extra space to open up). Examples below to illustrate this. Also, there's a special case for triangles that have the diameter of a circle as their base. Because the diameter is a 180 degree central angle, the angle opposite on the circumference of the circle must be 90 degrees. Not relevant for this question but good to know.

In the figure above, equilateral triangle ABC is inscribed in the circle.

How much of the circle is arc ABC? It's covering the same part of the circumference as two of the 60 degree angles from triangle ABA. Each of those angles represent 1/3 of the circle so arc ABC is 2/3 of the circle. If 2/3 is 24, 1/3 of the circle's circumference is 12, and the entire circumference is 36. So Pi*diameter = circumference. Pi*D = 36. D = 36/Pi. D = 36/3.14. A little less than 12. 11 is by far the closest answer choice. A little rusty feeling on these GMAT geometry rules? The rules are the easy part of GMAT studying. Get your fundamentals in shape ASAP. Avoid churning through questions before getting your basics GMAT quant organized. You don't want to waste good material and it's much harder to focus on the basics while working on tougher/trickier GMAT questions. Comment with any questions or additions. Happy studies!

In the figure above equilateral triangle ABC is inscribed in the circle. If the length of arc ABC is 24, what is the approximate diameter of the circle?

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