In the triangle above, does a^2 + b^2 = c^2 ?

Statement A:

if x+y=90. that means other side will be 90. this wud be right angle.

I think this is sufficient. A.

Statement B : x=y that means other side can be any number.

x=40 and y=40 other side cud be 100. so its not sure it cud be a2+b2=c2 or not.

so b is not sufficient.

Hence ans is a.

Statement 1) If x+y = 90 then the angle opposite of Side c has to have angle of 180-90 = 90, and the sides will satisfy the pythagorus theorem. Sufficient.
Statement 2) x = y. it is just a condition of isosceles triangle. Not Sufficient.

hence Option A)

Target question: Does a² + b² = c²?
This is a good candidate for rephrasing the target question.
a² + b² = c² should look familiar - it's the Pythagorean Theorem
In order for a² + b² = c², the triangle MUST be a RIGHT TRIANGLE
So, we can REPHRASE the target question....
REPHRASED target question: Is the triangle a right triangle?

NOTE: We can rephrase this version even more if we recognize that in the Pythagorean Theorem (a² + b² = c²), c represents the length of the hypotenuse.
So, in order for the Pythagorean Theorem to hold true in this example, the angle opposite side c must be a right angle.
So, we can RE-REPHRASE the target question as....
RE-REPHRASED target question: Is the angle opposite side c a right angle?

Now onto the statements!!!

Statement 1: x + y = 90
Since all 3 angles in a triangle must add to 180°, the missing angle must equal 90°.
In other words, the angle opposite side c IS a right angle
Since we can answer the RE-REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: x = y
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 45° and y = 45°. This means the 3rd angle (the angle opposite side c) = 90°. In this case, the answer to the RE-REPHRASED target question is YES, the angle opposite side c IS a right angle
Case b: x = 40° and y = 40°. This means the 3rd angle (the angle opposite side c) = 100°. In this case, the answer to the RE-REPHRASED target question is NO, the angle opposite side c is NOT a right angle
Since we cannot answer the RE-REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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Solution:

Question Stem Analysis:

We need to determine whether a^2 + b^2 = c^2. That is, we need to determine whether the triangle is a right triangle.

Statement One Alone:

Since x + y = 90, the unlabeled angle must be 180 - 90 = 90 degrees. In other words, it’s a right angle, and so the triangle is a right triangle. Therefore, we do have a^2 + b^2 = c^2. Statement one alone is sufficient.

Statement Two Alone:

Statement two is not sufficient. For example, if x = y = 45, then the triangle is a right triangle. However, if x = y ≠ 45, then the triangle is not a right triangle.

Answer: A

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