Question: 3 Triangles

Comment on 3 Triangles

how other angle is 60 degrees?
greenlight-admin's picture

Once we know the other 2 angles are 30 degrees and 90 degrees, we use the fact that all 3 angles must add to 180 degrees. For more, see

I thought when you draw an altitude, such as BD, from the vertex of a triangle you split the angle into two equal angles. So, for 105 I thought it would be 52.5.
greenlight-admin's picture

That rule applies only when the triangle is either an equilateral triangle or an isosceles triangle.
When dealing with an isosceles triangle, the rule applies when the two equal angles are at the base of the triangle, and we draw the altitude from the top vertex.

Thanks so much!

I thought, 6 should be equal to x sqrt(2)

6=x sqrt(2)
x = 3 sqrt(2)

then BD = 3, then BC = 6.

what is the mistake i have done?

I'm sure this is incredibly late, but maybe it'll help you or somebody else if they have the same problem.

For 45-45-90 triangles, we know that the ratios for the sides are 1:1:√2

If our 90 degree angle has an opposite side length of 6, we know that x(√2) = 6.
Solving for x, you divide by √2 and get an enlargement factor of 6/√2
Conversely, you can solve for x and get 3√2. This works because if you multiple √2 by √2, you get √4, which simplifies to 2. You can then multiply 2 by 3 to get 6.
greenlight-admin's picture

Sorry, I have no idea how this question slipped by me!

You have successfully determined that the ENLARGEMENT FACTOR (for BOTH triangles) = 3√2

Side BC (with length x) is part of a 30-60-90 triangle.
The BASE 30-60-90 triangles has lengths 1, √3, 2
Side BC corresponds with the side with length 2 (in the base triangle).
To find the length of side BC, we apply the enlargement factor of 3√2

That is, x = (2)(3√2) = 6√2


how do we know that it is a 45degree angle for A and B vertices.even though it is isosceles?
greenlight-admin's picture

Since AD = BD, we know that ∆ABC is ISOSCELES
This means that ∠DBA = ∠DAB.
Let's let x = ∠DBA
This also means x = ∠DAB

We also know that ∠BDA = 90°

Since all three angles in ∆ABC must add to 180°, we can write:
x + x + 90° = 180°

When we solve this equation, we get x = 45°

In other words, ∠DBA = ∠DAB = 45°

Does that help?


Awesome thnx.


Great website. One question in regards to finding the largest side of the 30:60:90 triangle BDC as 6 root 2. If I'm not mistaken, that is also the length of the smallest size. How can the hypotenuse share the 6 root 2 side length with the x side.
greenlight-admin's picture

Hi jameconn98,

Sorry but I'm not entirely sure what you mean by "...that is also the length of the smallest side."

BD is the shortest length of the 30-60-90 triangle BDC, and BC (aka x) is the longest length of triangle BDC.

BD = 6/(√2) whereas BC = 6√2

Does that help?


Hi Brent,

Sorry about that. I misread the text on the video.

greenlight-admin's picture

No problem. It's always a good idea to ask if anything seems off.


Have a question about this video?

Post your question in the Comment section below, and a GRE expert will answer it as fast as humanly possible.

Change Playback Speed

You have the option of watching the videos at various speeds (25% faster, 50% faster, etc). To change the playback speed, click the settings icon on the right side of the video status bar.

Let me Know

Have a suggestion to make the course even better? Email us today!

Free “Question of the Day” emails!