Lesson: Triangles - Part II

Comment on Triangles - Part II

How could we assume the height is 3 & 7.5 ? Is there a formula to find the height?
greenlight-admin's picture

You're referring to the triangles at 4:25 and 5:40 in the video.
Those heights are just hypothetical (in fact, they aren't accurate at all). The sole purpose of the video is to explain how to find the area of a triangle using the formula (base)(height)/2

In some cases, there will be additional information to help us determine the height.


In the above question, how did you calculate the hight?

I taught third side must be between 2 and 14, but with do we have any relation from which we can know the hight of triangle?
greenlight-admin's picture

Link: http://greprepclub.com/forum/two-sides-of-a-triangle-have-length-6-and-2...

The two known sides have lengths 6 and 8. So, if we make the side with length 8 the base, then the biggest height possible is when the side with length 6 is perpendicular to the side with length 8 (as shown in my first diagram).

At this point, all I need to do is keep increasing the angle beyond 90 degrees until it has a height of 3.

How do we know this is possible?

Well, we know that we can certainly create a triangle with height 6, and we know that, if we keep increasing the angle between the two known sides, we can make the height of the triangle less and less and less, until the height approaches zero.

So, we can say the following about POSSIBLE heights:

0 < height ≤ 6 (if we make the base the side with length 8)

Hi Brent. If in the GRE exam some problem doesn't haven any information of the triangle high, and also I is no possible to apply Pythagorean theorem. How I could find the area? Thanks.
greenlight-admin's picture

If the height is not given and you cannot apply the Pythagorean theorem, then you COULD use something called Heron's Formula (http://www.mathwarehouse.com/geometry/triangles/area/herons-formula-tria...).

However, this formula is out of scope for the GRE. That is, you are not expected to know it for the test.

Hi Brent,

In the following question if it were mentioned that the line drawn from top vertex is altitude, can we say then x=y since the triangle is isosceles.

greenlight-admin's picture

Question link: http://greprepclub.com/forum/x-or-y-is-greater-3611.html

That's right. If it were the case that the horizontal line is the altitude (i.e., it creates a 90-degree angle with the base), then we could conclude that x = y.

My exam is on the 22nd what are the areas i should concentrate on in quants and how often do questions between the difficulty level 160-170 appear on the test
greenlight-admin's picture

The test is computer adaptive. So, the number of 160 to 170-level questions you encounter will depend on how well you performed on the section before that.

For example, if you correctly answered 2 of the 20 questions in the first quant section, then your second quant section would likely consist of easy and perhaps some medium difficulty questions.

For more on the computer adaptive nature of the GRE, start watching at 1:40 of the following video: https://www.greenlighttestprep.com/module/general-gre-info-and-strategie...

If you have limited time to prepare, I suggest that you focus on the concepts that have the greatest return on investment (ROI). These are concepts that are tested frequently and require the least amount of time to learn. In my opinion, these concepts are:

- Quantitative Comparison strategies
- Percents
- Ratios
- Integer Properties
- Powers/roots
- Algebra
- Geometry
- Statistics

Finally, before your test, you should take AT LEAST one full-length (official) practice test (https://www.ets.org/gre/revised_general/prepare/powerprep/?WT.ac=gre_362...) so you can familiarize yourself with the interface AND work on your timing and endurance.


how did we assume PQR is 90deg?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/squares-pqrv-and-vrst-have-sides-of-length...

We're told that PQRV is a square.
So, ∠PQR = 90° (as do all 4 angles in PQRV)



Can you please explain this again
greenlight-admin's picture


I have a step-by-step solution here: https://greprepclub.com/forum/what-is-the-area-of-a-triangle-created-by-...

Can you tell me which parts of that solution are giving you difficulties?



For this question, I was working with an assumption that did not work out in the end.

Since angle C = z, B = z and A = 2z I was assuming that the opposite side of A(BC) should be 2 times the opposite sides of B and C (which are both of length 1 in this case). I was making this assumption since angles in a triangle correspond with opposite sides. Could you please explain what is wrong with my assumption?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/in-the-figure-what-is-the-area-of-abc-1107...

That type of relationship (between angle measures and the lengths of their opposite sides) does not exist.

Take, for example, the 30-60-90 right triangle.
Its sides are 1, 2 and √3.
- the side opposite the 30° angle has length 1
- the side opposite the 60° angle has length √3
- the side opposite the 90° angle has length 2
Notice that the side opposite the 60° angle is not twice the side opposite the 30°


Hi Brent,

Do the" isosceles triangle " need to have all 3 sides are the same length?
greenlight-admin's picture

From the Official Guide:

An isosceles triangle has AT LEAST two sides of the same length.
An equilateral triangle has ALL sides of equal length.

So, we can say that an equilateral triangle is a type of isosceles triangle.

Does that help?


Hi Brent,

Can you help me with this question?
If you can provide the solution, that would be appropriate.

It seemed like this question needed sin/cos knowledge.
greenlight-admin's picture

The GRE would never have a question that required a knowledge of sine, cosine, etc

Here's my full solution: https://greprepclub.com/forum/dbc-is-an-equilateral-triangle-abd-is-an-i...


That meant I need to remember 30-60-90 right triangle (2,1 and √3).
am I correct? How about other forms traiangle? (like 45-45-90).
do I need to remember for GRE too?
greenlight-admin's picture

Yes, you need to remember the 30-60-90 triangle and the 45-45-90 triangle.


In the following question:

Area of an equilateral triangle = (side²)(√3/4)

Is that a formula ? Or from where we get root of 3/4

Thank you

greenlight-admin's picture

Link: https://greprepclub.com/forum/if-an-equilateral-triangle-has-an-area-equ...

I should have added brackets to avoid ambiguity. The formula is:

Area of an equilateral triangle = (side²)[(√3)/4]
The formula is discussed in the above video.

The formula is derived by taking an equilateral triangle, and breaking it into two right triangles by drawing the altitude of the triangle. Doing so divides the equilateral triangle into two 30-60-90 right triangles and then using the ratios of this special triangle to calculate the area.

Here's a short proof if you're interested: https://www.youtube.com/watch?v=pdKcNbKW8i0



I am looking at your solution to this question:

So we are given the two lengths of the triangle: 6 and 8. In your solution, you maximize the triangle's area by doing 6(8)/2, but what about the rule that says when you add the two legs together, the third side must be less than that? So isn't the max height closer to 14? Also, same thing with the minimum. Doesn't the shortest side need to be more than 2?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/two-sides-of-a-triangle-have-lengths-6-and...

You are correct to say that: 2 < length of third side < 14
However, the goal here is to find the range of possible AREAS (not possible LENGTHS of the third side).

If we make the side with length 8 the base, then the are you will be maximized when the height is maximized.
So, as you can see from the diagram in my solution, greatest possible height must be 6.

Even though the length of the third side can be greater than 6, this does not mean that the height of the triangle can be greater than 6.
For example, if you try to create a triangle such that the 3rd side has length 13 (which is pretty close to 14), the resulting 6-8-13 triangle will be very flat. More importantly, the height of the triangle will be very very small.
See: https://imgur.com/QgxyFat

Does that help?

It sort of makes sense to me! I think I will understand it in time if I just think about it more and do more problems.

Thank you!

Hey Brent! Question on this problem: https://greprepclub.com/forum/comparison-ab-or-bc-11822.html

Why can't we use the rule that when two angles are equal, the sides opposite to the two angles would also be equal and consider AB = BC?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/comparison-ab-or-bc-11822.html

If a triangle (a single triangle) has two equal angles, then that triangle is an isosceles triangle, which means that has two sides of equal length.

However, in the question above, we don't have a single triangle with two equal angles; we have triangle ABC with one 30° angle, and we have triangle DBC with one 30° angle. As such, we can't make any conclusions about equal sides.


Hi Brent!
Is there a theorem or a prof that area of right-angled triangle is maximum. I wanted to generalize this question
greenlight-admin's picture

Question link: https://greprepclub.com/forum/two-sides-of-a-triangle-have-length-6-and-...

If we're given two line segments comprising two sides of a triangle, the maximum area is obtained when the two given sides are perpendicular.
Here is a video proof involving calculus: https://www.youtube.com/watch?v=SWJmXJ_1iyA

Perfect ! 1/2 ab sin(theta) answers the question

https://greprepclub.com/forum/what-is-the-area-of-quadrilateral-abcd-12299.html. What if draw a line from angle B to D The Area is then coming 12? and overall area 42. Can you point me where I am wrong?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/what-is-the-area-of-quadrilateral-abcd-122...

If we draw a line connecting B and D, we get: https://imgur.com/o6T2T65
As you can see, the area of triangle ABD = 12, but there's no easy way to find the area of triangle BCD.

How did you conclude that the area of triangle BCD = 30? (to get a combined area of 42)

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