What is the area of the triangle with sides a, b, and c?

Question

What is the area of the triangle with angles a, b, and c?

(1) a = x, b = 2x, c = 3x
(2) The side opposite a is 4 and the side opposite b is 3.

Source: Nova's GMAT prep course, page 298, exercise 7)

IanStewart · Answer

The question doesn't make sense. A triangle cannot have the side lengths provided in Statement 1, because the sum of any two sides of a triangle must always exceed the third side. But x + 2x is not greater than 3x. So from Statement 1 the triangle cannot exist. Statements 1 and 2 are also inconsistent; they can't both be true. So if the question has been reproduced correctly, there are several things deeply wrong with it.

Xin Cho · Answer

My apologies, I have noticed I have incorrectly copied the example from the book. I have attached actual picture of the prompt from the book.

Trev50 · Answer

Statement 1: Only having angles isn't sufficient to find the area because of similar triangles.
Insufficient

Statement 2 : Area = 0.5 * base * height = 0.5 * 4 * 3 =6
Sufficient

Ans should be b then

Posted from my mobile device

Xin Cho · Answer

In the statement 2, you are assuming that angle c is 90 degree angle. But it might be 89 degrees or 91 degrees - we don't know.
For that reason statement 2 alone is insufficient.

Taking both statements together:
I) sum of angles is x+2x+3x = 180 --> x=30 --> thus, we know that angle c is a right angle.
II) side a = 4, side b = 3
Thus, the area is 0.5*3*4=6

Staphyk · Answer

Answer C

IanStewart · Answer

The question still makes no sense. If you use both Statements, for one thing, a is the smallest angle of the triangle, so it must be opposite the shortest side. But according to Statement 2, it's not opposite the shortest side. And for another thing, using Statement 1, we must have a 30-60-90 triangle. The sides of a 30-60-90 triangle are in a 1 to √3 to 2 ratio. So it's impossible for one side to be 3 and another to be 4, as Statement 2 claims. So the two statements make no sense when you consider them together, since they're describing a triangle that can't exist.

Xin Cho · Answer

Aha, haven't actually thought of that - also can't believe that it is wrong in the book. Thank you very much for checking!

hovhannesmkrtchyan · Answer

Although I marked the "correct" choice, I have noticed something strange about this question, so I decided to share it with everyone.

The question asks the value of area of the triangle.

So, per statement one:
(1) We can get the value of each of the angles by x+2x+3x=180 and we get x=180/6=30 degrees. 
So we have a 30,60,90 degrees triangle. But, as we don't have any info about the lengths of sides we can't calculate the area of triangle.

Now, statement two tells us:
(2) that two sides of the triangle are 4 and 3, and the only thing that we can infer is that the third side ranges from 2 to 6.
Since we don't have any info about the type of triangle and about the height of the triangle we can't calculate the area.

So, combining statements one and two:
(1)+(2) we know that the opposite of side length of a is 4 and the opposite of side length is 3, and the biggest angle in this traingle is angle c, which is 90 degrees, we can infer that 3 and 4 are the legs of right triangle, so the area of triangle will be 3*4/2=6.

However, since we know that we have a 30,60,90 triangle and we also know that the side opposite a is 4 and the side opposite b is 3, and therefore the side
opposite to c must be 5, we here violate some of the basic properties of 30,60,90 triangle, which is the opposite side of 30 degree angle is equal to half of the hypotenuse (in this case the hypotenuse can't be 5, it should have been 6). However this can't be true because now 3^2+4^2 = 6^2, which is obviously false.
That's why I believe that the statements one and two are inconsistent. So, in reality you can't find the area of the figure which is can't exist.

I think one way to correct this question is to assign the opposite of a to 3√3, and then we will have 30,60,90 triangle with sides 3,3√3 and 6 respectively. Though the area will be 3*√3/2, but it's not really important. All that matters is that the statements are consistent now, and question makes sense.

What is the area of the triangle with sides a, b, and c?

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What is the area of the triangle with sides a, b, and c?

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