In the figure above, what is the value of z?

IMPORTANT: For geometry Data Sufficiency questions, we are typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in the video at the bottom of the page.

Target question: What is the value of z?

Statement 1: x = y = 1
This statement locks a few lengths into place. However, if we MENTALLY grab the bottom right vertex, we can pull it right to left, without affecting the fact that x = y = 1.
This means that we can alter the length of bottom side, which means we can alter the value of z
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: w = 2
This statement locks the hypotenuse into place. However, if we MENTALLY grab the left-most side (with length x), we can move that side right and left, without affecting the fact that w = 2
This means that we can alter the length of bottom side, which means we can alter the value of z
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Knowing that x = 1 and w = 2 means we can use the Pythagorean Theorem to find the length of the bottom side of the right triangle.
Once we know the length of the bottom side of the right triangle, we can add it to 1 to get the length of bottom side, which means we can definitely determine the value of z
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent



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We need to determine the value of z in the following diagram:



Note that z is equal to the length of the rectangle and the base of the triangle. If we let the base of the triangle = b, z = b + y.

Statement One Alone:

x = y = 1

Although we know that x = 1, we do not know the value of b or the base of the triangle. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

w = 2

Knowing only the value of w does not allow us to determine z. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using the information from statements one and two, we know that x = y = 1 and w = 2, and we can determine b using the Pythagorean theorem.

x^2 + b^2 = w^2

1^2 + b^2 = 2^2

1 + b^2 = 4

b^2 = 3

b = √3

Thus, z = y + b = 1 + √3.

Answer: C
_________________

Stat 1: X = Y = 1 , from the figure we can understand that all sides are equal and it is square. Z = 1 + A .. we are not sure of A ( A is the bottom line in right angled triangle)...Insufficient.

Stat 2: W = 2... We are not sure the about atleast the other side of the right angled triangle to know the value of A...Insufficient.

Stats 1 + 2 = \(1^2\) + \(A^2\) = \(2^2\) ... we can get the value of A... both sufficient to get the value of Z.

Hence C is correct answer..

C

I got mislead by reading 2 as sq.root of 2, and ended up with A ..
(thinking that B is just another way of writing A).

C is good.

x=y=1 implies its a square but we still need one ore side of the trianglee or an angle to find the other side with first statement so insufficient.
Statement 2 is clearly insufficient alone.

But both the statements combined are sufficient as we have 2 sides known.

Answer: Option C

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Hi, I need someone's help clarifying one thing.

I read somewhere that in GMAT dont assume anything that is not mentioned and in this question it is not mentioned that opposite sides of the quadrilateral are equal so in my head I can't assume it as a square/rectangle and thus wont be able to assume second side of triangle as 1 and thus cant apply pythagoras theorem. I see OA is C so my that understanding is definitely incorrect.

Can somewhere clarify what level of assumptions can be made and cannot be made especially in geometry questions where you have figure and can see it to be a square/rectangle like in this case