Lesson: Triangles - Part I

Comment on Triangles - Part I

For these difficult questions with the answer choice 'E. Cannot be determined from the information above' - can I assume that for the most part these are 'trap' answer choices? (i.e. test makers know it's not easy so test takers are compelled to say that it cannot be determined using the info given)
gmat-admin's picture

I'm assuming you're referring to Data Sufficiency (DS) questions.

It's true that, for the very hard DS questions, it's more likely that the correct answer is A, B or D. HOWEVER, the problem with this strategy is that the computer adaptive nature of the GMAT ensures that everyone struggles. So, most people may feel that the correct answer to their tricky question will also be more likely to be A, B or D.

sir one more
https://gmatclub.com/forum/if-the-sides-of-a-triangle-have-lengths-x-y-and-z-x-y-113217.html

Hi Brent,

Could you please explain solution to this answer. I am not able to figure it out?
https://gmatclub.com/forum/in-the-figure-above-what-is-the-value-of-x-y-144330.html

Also, Are Similar & Identical triangles one & the same or do they differ? Also, in case of two identical triangles, are their sides equal or are they equal in proportion?
gmat-admin's picture

Question link: https://gmatclub.com/forum/in-the-figure-above-what-is-the-value-of-x-y-...

You bet!
Here's my solution: https://gmatclub.com/forum/in-the-figure-above-what-is-the-value-of-x-y-...

ASIDE: Identical triangles are . . . identical. Their corresponding angles are all equal AND their corresponding sides have equal length.

In SIMILAR triangles, the corresponding angles are equal, and the corresponding sides are PROPORTIONAL to each other.

Here's the video on similar triangles: https://www.gmatprepnow.com/module/gmat-geometry/video/872

https://gmatclub.com/forum/in-triangle-abc-above-what-is-the-length-of-side-bc-168281.html

Do you have a video explanation on this question? I can't seem to understand why statement 1 is sufficient.

Here's my approach:

The information I got from the from the question alone:

For triangle ABC angle ABC = 180 - 3x
Triangle BDC is isoceles with lines BD = CB
Angle DBC = 180 - 4x
Angle ABD = 180 - sum of angles BAD and angles BDA (x + 180 - 2x) = 180 - x
Similarly, Angle DBC would equal 180 - 4x

As such we know that AB = AD and BD = BC
AD is NOT equal to BC (they have different angles to begin with)

So where have I gone wrong?
gmat-admin's picture

Here's my full solution: https://gmatclub.com/forum/in-triangle-abc-above-what-is-the-length-of-s...

Aside: You wrote "AD is NOT equal to BC (they have different angles to begin with)"
Just because sides AD and BC belong to different-looking triangles, doesn't mean they can't have the same length.

Cheers,
Brent

https://gmatclub.com/forum/in-the-figure-shown-above-x-is-the-length-of-side-bd-of-triangle-abd-213796.html hi Brent. Isn't the big triangle ABC isosceles?
So if lengths opposite same angle are same, wouldn't the answer for BD be 4? Cos BC and BD are opposite same angle B ...
gmat-admin's picture

Question link: https://gmatclub.com/forum/in-the-figure-shown-above-x-is-the-length-of-...

Be careful; you are concluding that ∆BDC is isosceles, but this is not necessarily true.

Yes, ∆ABC is isosceles, because AC = BC = 4
However, this does not mean that doesn't mean that ∆BDC is isosceles.
To show that ∆BDC is isosceles, we need to show that it has two equal angles and/or two equal sides.

In fact, side BD cannot be 4, since that would break the rule that says, "If two sides of a triangle have lengths A and B, then:
(DIFFERENCE between A and B) < length of third side < (SUM of A and B)

Cheers,
Brent

Hi Brent,

What are your thought on the question :- https://gmatclub.com/forum/if-abd-is-a-triangle-is-triangle-abc-a-right-?
As per me the answer should be E as B can lie anywhere ?
gmat-admin's picture

Hi Brent,

Could you please have a look at the following question?

https://gmatclub.com/forum/a-triangle-with-three-equal-sides-is-inscribed-inside-a-160874.html#p2464632

I understand how the probability was derived, but this step that Bunuel has suggested is confusing me a bit:

For equilateral triangle the radius of the circumscribed circle is R=side∗3√3R=side∗33, thus the area of that circle is πR2=π∗side23πR2=π∗side23 .
Are we expected to know this formula or there is a way to get to this?

Thanks
gmat-admin's picture

Question link: https://gmatclub.com/forum/a-triangle-with-three-equal-sides-is-inscribe...
I'm not a big fan of memorizing tons of formulas.
If we recognize that an equilateral triangle has some 30-60-90 special right triangles hiding in it, we can always determine the area of the inscribed equilateral triangle.
For example, if we have a circle with radius 2, we got the following scenario: https://imgur.com/1AYhFi6
Once we determine the lengths of one hidden 30-60-90 special right triangle, we can quickly find the area of the entire equilateral triangle.

Here's a question where I've added a few extra steps: https://gmatclub.com/forum/what-is-the-perimeter-of-an-equilateral-trian...

Thanks Brent, so for questions like these can I assume that the radius of the circle is say for example 10 (whatever value I want even if the question doesn't give any specific values for the length of the radius) and then try to derive the other sides of the right triangle, which in turn can give me all sides of the equilateral triangle?
gmat-admin's picture

That's correct. Since the question (https://gmatclub.com/forum/a-triangle-with-three-equal-sides-is-inscribe...) requires us to determine the RATIO of the triangle's area to the circle's area, we can assign any value we want to the radius, and the answer will be the same no matter what.

Got it, thank you Brent :)

Also, can you please give a general tip for when assigning values to solve the questions is best? Like different scenarios where it's most common and most practical way to get to the solution?

Thank you!!
gmat-admin's picture

As a general rule, we can assigned values when a geometry question asks us to find some proportion (e.g., ratio, fraction, percent).
In this case, the proportion is disguised as a probability.
But that is, we want to find the fraction: (area of inscribed triangle)/(area of circle)

Thanks Brent,

Can you please also explain the following question?

https://gmatclub.com/forum/in-the-diagram-above-pqr-is-a-right-angle-and-qs-is-131991.html#p1082335

I understand why the 3 triangles are similar, however, in terms of the relation between the sides of the triangles, I would assume PS/QS=SR/QS?

Thanks
gmat-admin's picture

Determining corresponding sides can be tricky.
Please see my full solution: https://gmatclub.com/forum/in-the-diagram-above-pqr-is-a-right-angle-and...

Hey Brent, do you have a solution for this? Looked through GMAT Club and none really resonate. I now understand why 2 is insufficient, which I originally thought was sufficient. But cant quite understand why 1 i sufficient. Seems like the small triangle, although a set perimeter, the sides can vary in length.
https://gmatclub.com/forum/in-the-figure-above-pqr-and-stu-are-identical-equilateral-triangles-207708.html
gmat-admin's picture

Very tricky question!!
Here's my full solution: https://gmatclub.com/forum/in-the-figure-above-pqr-and-stu-are-identical...


https://gmatclub.com/forum/is-the-perimeter-of-square-s-greater-than-the-perimeter-of-e-167420.html

Hi Brent,

Can you please help me with the solution to above question
gmat-admin's picture

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