If the circle has radius 6, what is the area of the triangle?
(1) AC = AB
(2) BC = 12
Circle and triangle
To calculate the area of a triangle, we use the formula,
So, we need to be able to determine the triangle's base and height.
We are told that two of
's sides are equal. This makes it an isosceles triangle. However, this tells us nothing about the actual lengths of the sides, except that sides AB and AC are equal.
It’s important to notice that we don’t know anything about side BC. It looks like it goes through the middle of the circle, but we cannot just assume that it does. So we don’t know if BC is a diameter or not, and we don’t know it’s length.
Thus, we don’t know have enough information to determine the base and height of
. So statement 1 alone is not sufficient.
This gives us the length of BC. Since it is 12, and the radius is 6, BC must be the diameter. However, we know nothing about AB or AC.
We can draw another conclusion from this:
We know this because any inscribed angle that intercepts the endpoints of a diameter, will be a
angle. This rule is explained here, about 7:00 minutes into the video.
By the way, if you aren’t sure how we know that point A is on the edge of the circle, check out the FAQ below!
So BC = 12 and BAC =
. We could use BC as the base of our triangle, but we would have no height! Because we were given no information about AB or AC, our triangle could still take a variety of shapes (see image below). Statement 2 alone is not sufficient.
Statements 1 & 2:
Together, these statements tell us that we have an isosceles right triangle with a hypotenuse of 12. Now we can use the Pythagorean Theorem to calculate the legs.
Let’s label AC and BC as x. We have:
x2 + x2 = 122
2x2 = 144
x2 = 72
Since this is a right triangle, we can use x as both our base and our height. Thus, we can calculate the area.
BOTH statements 1 and 2 TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
Still have a question? Click here for even more FAQs.
Frequently Asked Questions (FAQs)
Q: Since the prompt tells us that the radius is 6, don't we know that BC is the diameter without using Statement 2?
A: Notice what you've done: you've assumed that BC is the diameter! But we're not actually told that, and there's actually nothing in the picture that indicates that this is the case. Sure, the line looks like it could be the diameter, but there's nothing marking it as such. Therefore, based on Statement 1 alone, we absolutely CANNOT just assume that it is the diameter. We have to have some kind of evidence to point us in that direction.
It's only when we have statement 2) that we can conclude BC is the diameter because it's exactly twice the radius.
This lesson video discusses assumptions you can and cannot make regarding GMAT diagrams.
Q: Since AC=CB in Statement 1, can we conclude that ∠BAC is is 90º and that the triangle is a right triangle?
A: No, this does not mean that BAC is a right triangle. The act of being inscribed in a circle does not alone guarantee that the inscribed triangle is a right triangle. However, there is one way to guarantee for sure that an inscribed triangle is a right triangle: if the triangle's hypotenuse is the circle's diameter, then the triangle is a right triangle, with the right angle opposite the diameter.
As mentioned in the FAQ above, although BC looks like the diameter, we don't know this for sure without the information from Statement 2!
Q: How do we know that point A in
is on the edge of the circle?
A: Keep in mind that for GMAT diagrams, if it looks like a certain point is on a given line, we can assume that to be the case. So since it looks like point A is on the edge of the circle, we can assume that that’s the case!
Watch the lessons below for more detailed explanations of the concepts tested in this question. And don't worry, you'll be able to return to this answer from the lesson page.