Is Triangle ABC an Isosceles? GMAT Data Sufficiency

Question:

Is triangle ABC an isosceles?

(1) AB/BC = 2
(2) x≠y

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: C

Solution and Explanation:
Approach Solution 1:
Given both assumptions, the triangle can only be isosceles if BC = AC, which isn't feasible because side AC + BC (if they're the same length) equals AB.
In addition, the triangle's third side must be more than the total of the other two sides or less than the difference between the two sides.
Therefore, it can be said with certainty that the triangle is not isosceles if it does not meet both conditions.

Thus explaining further ahead:
Statement 1 alone: It is given that, AB/BC = 2
Therefore, AB≠BC.
Hence, statement one is Not sufficient on its own.

Statement 2 alone: It is given that x ≠ y
So, AB≠AC.
Therefore, statement two is not sufficient on its own.

Combining both statements we get:
From both statements, we know that AB≠BC and AB≠AC.
Therefore, the only possible way in which the ABC triangle can be isosceles is when AC=BC.

However, in this case, since it is already given in statment one that AB=2BC
then AB = BC + BC = BC + AC
But this case cannot be correct since in an isosceles triangle the length of any side of a triangle must be smaller than the total of the other two sides or both sides.
Therefore, AC≠BC,
Therefore, the ABC triangle is not isosceles.

Hence, both statements together are sufficient.

Approach Solution 2:
The problem statement asks to find if ABC is an isosceles triangle.

As per the rules of an isosceles triangle, two sides and angles should be equal. Moreover, the third side of the triangle must be more than the total of the other two sides or less than the difference between the two sides.

Statement 1: AB/BC = 2
This statement implies that angle A = 2Y
However, we cannot certainly confirm whether this is an isosceles triangle because
2(50) + 50 + x =180
x = 30
Or it could be isosceles if
2(45) + 45 +45 =180
Therefore, we cannot ensure that whether ABC is an isosceles triangle unless we have more information.
Hence, statement one alone is Insufficient.

Statement 2: x≠y
This statement also does not confirm that whether ABC is an isosceles triangle or not.

Combining Statement 1 and Statement 2, we get:
If x cannot equal y then we cannot have an equilateral.

Therefore, the ABC triangle is not isosceles.

Hence, both statements together are sufficient.

Approach Solution 3:
The problem statement asks to find if ABC is an isosceles triangle.
Let's consider angle A is z.
Statement 1: AB/BC = 2
This implies y = 2z

There is no information regarding whether x = y. Therefore, this statement alone is insufficient.

Statement 2: x≠y
This implies x cannot equal y
There is no information about the fact that whether x = z. Therefore, this statement alone is insufficient.

Combining both statements, we get:
The 3 angles of the triangle - x, y, and z, where each of which is distinct.

Hence, ABC is NOT an isosceles triangle.
Therefore, both statements together are sufficient.

“Is triangle ABC an isosceles”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions are featured with a problem statement that is followed by two factual statements. GMAT data sufficiency includes 15 questions which are two-fifths of the total 31 GMAT quant questions.

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