In triangle ABC (I added the triangle as an image), what is

[Guest]

In triangle ABC (I added the triangle as an image), what is the length of side BC?

1) Line segment AD has length 6
2) X = 36



The answer is A.
I understand that triangle BCD is isosceles and I was pretty sure that statement 2 was not enough information. How does statement 1 provide enough information to answer this question. I wasn't clear on how to use the statements.

[StaceyKoprince]

Hi, I'm getting an error when trying to view the image. Can you either upload again or try to explain the image via text?

[Guest]

There is a large triangle ABC. From vertex B (on the top of the image) is a line (BD) to the middle'ish of the other side (AC). Angle A is represented by x degrees. Angles BDC and BCD are each 2x degrees. I hope this helps.

[StaceyKoprince]

Basically, you have to follow through a bunch of steps from one side of the triangle to the other. This is what I call a GMAT Proof problem - they use data sufficiency sometimes to write questions that seem somewhat proof-like.

So two of the angles are labeled 2x. That means the corresponding sides (BD and BC) are the same length. Draw little slashes on those lines to indicate they are congruent.

To the left of the middle 2x (by D), write 180-2x (because those two angles together add up to a straight line = 180). On the smaller of the two angles by letter B, write "y."

x + y + (180-2x) = 180 (because it's a triangle). The 180's cancel out and we have x + y - 2x = 0, or y-x = 0 or y = x. On your diagram, replace the angle labeled "y" with an x. **

Now, two angles in the smaller triangle are both labeled x, so the corresponding sides are also equal. Those sides are AD and BD. BD was also part of the earlier pair of congruent sides, so now we know that AD = BD = BC, and AD 6 (from statement 1). Sufficient.

** Slight shortcut here for next time. Know that the middle 2x is what's called an exterior angle of a triangle. An exterior angle is always equal to the sum of the two opposing interior angles (in this case, x and what we originally labeled y). In other words, we could have just said x + y = 2x, therefore x = y. But I wanted to show you the long way first to make sure you understood the actual proof behind it.

[condenach]

Hi,

As you said triangle BDC is isosceles, so if you pick numbers for X (let´s say 37º wich is a rare one, it works with every value you pick) then you have BDC=74º, BCD=74º and DBC=32º

Now if you move to triangle ADB you have that BAD=37º, ADB=180º-2X=180º-74º=106º and then you can get ABD by doing 180º-BAD-ADB= 180º-37º-106º=37º

So you discover that triangle ABD is also isosceles and then you have you answer because you have 2 isosceles triangles so side AD equals side BD in ADB and in triangle DBC side BD equals side BC, so BC is also 6.

Obiously knowing that x=36º it is not enough if you don´t have the measure of at least one side of the triangles.

Hope it helps

[Guest]

VERY helpful explanations. Thanks!