Data Sufficiency

ziyuenlau wrote:Is a+b>c?

1) a, b, and c represent three different lengths of the sides of a certain triangle

2) a^2+b^2=c^2

Source: Math Revolution
OA=A

Statement 1: a, b, and c represent three different lengths of the sides of a certain triangle.

For a triangle, the sum of any two sides is greater than the other side. Thus, a+b>c. Sufficient.

Statement 2: a^2+b^2=c^2

Reading the relationship, a^2+b^2=c^2, after Statement 1, one tends to think of ONLY Pythagorean theorem, making a conclusion that a and b are base and height, and c is the hypotenuse of a right-angled triangle. Thereby apply the rule that for a triangle, the sum of any two sides is greater than the other side. Thus, a+b>c.

However, the relationship, a^2+b^2=c^2, must not be viewed only from the perspective of a triangle. The question prompt does not state that a, b, and c are the three sides of a triangle.

Case 1: For a=3, b =4 and c =5, a^2+b^2=c^2, and a+b = 7 > c = 5. The answer is YES.

Case 2: For a=3, b =-4 and c =5, a^2+b^2=c^2, and a+b = -1 < c = 5. The answer is NO. No unique answer.

The correct answer: A

Hope this helps!

Relevant book: Manhattan Review GMAT Geometry Guide

-Jay
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ziyuenlau wrote:Is a+b > c?

1) a, b, and c represent three different lengths of the sides of a certain triangle

2) a² + b² = c²

Great question!

Target question: Is a+b > c?

Statement 1: a, b, and c represent three different lengths of the sides of a certain triangle
IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
DIFFERENCE between sides A and B < third side < SUM of sides A and B
So, we can be certain that a+b > c
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: a² + b² = c²
Statement 2 is what I love about this question!
Statement 1 got me thinking about triangles, then this equation (which looks exactly like the Pythagorean Theorem), got me thinking about right triangles. So, my initial reaction was to reapply the concept from statement 1 to conclude that statement 2 is also sufficient.
Then it dawned on me that, even though a² + b² = c² COULD apply to a right triangle (where the length of each side is a positive number), it could also apply to negative values of a, b, and c.
In fact, there are infinitely many values of a, b, and c that satisfy statement 2. Here are two:
Case a: a = 3, b = 4 and c = 5, in which case a+b > c
Case b: a = -3, b = -4 and c = 5, in which case a+b < c
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

hoppycat wrote:For statement 1 how do you know that c is the third side?

Sorry, I should have clarified that.
If a, b and c represent the 3 lengths of a triangle, we can use the rule to conclude ALL of the following:
|a - b| < c < a + b
|a - c| < b < a + c
|c - b| < a < c + b

Cheers,
Brent