The Angles in a Triangle are x, 3x, 5x Degrees GMAT Problem Solving

Question: The angles in a triangle are x, 3x, 5x degrees. If a, b, c are the lengths of the sides opposite to angles x, 3x, 5x respectively, then which of the following must be true?

1. I and III only
2. II and III only
3. I only
4. II only
5. III only

“The angles in a triangle are x, 3x, 5x degrees. If a, b, c are the lengths of the sides opposite to angles x, 3x, 5x respectively, then which of the following must be true?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

Approach Solution 1:

There is only one solution approach to this problem.

According to the relationship of the sides of a triangle: the length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.
Thus I and II can never be true: one side © cannot be larger than the sum of the other two sides (a and b)
Note that II is basically the same as I: if c = 10, a = 6 and b = 2 then c > a + b, which can never be true. Thus even not considering the angles, we can see that only answer choice E (III only) is left (all other options are out because each of them has either I or II in them).
Now if interested why III is true: as the angles in a triangle are x, 3x, 5x degrees then x + 3x + 5x = 180.

Hence x = 20, 3x = 60 and 5x = 100.
Next if angle opposite side c were 90 degrees, then according to the Pythagoras theorem \(c^2=a^2+b^2\) but since the angle opposite side c is more than 90 degrees (100) then side c is larger, hence \(c^2>a^2+b^2\)

Correct Answer: E

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