Data Sufficiency

jjjinapinch wrote:

In the figure above, what is the value of z?
1) x = y = 1
2) w = 2

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

In the figure above, what is the value of z?
1) x = y = 1
2) w = 2

Official Guide question
Answer: C

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We wish to get the value of z. We see that z = length of the quadrilateral + Base of the rightangled triangle

We see that the three angles of the quadrilateral are 90 degrees each, thus, the fourth one would also be 90, making the quadrilateral a rectangle.

Thus, the length of the quadrilateral (rectangle) = y

and the height of the rightangled triangle = x

Statement 1: x = y = 1

=> The length of the quadrilateral (rectangle) = y = 1

and the height of the rightangled triangle = x = 1

But we cannot get the value of the base of the rightangled triangle. Insufficient.

Statement 2: z = 2

We do not know the length of the quadrilateral (rectangle). Insufficient.

Statement 1 and 2:

We know that the length of the quadrilateral (rectangle) = y = 1

We know that the height of the rightangled triangle = x = 1 and the hypotenuse of the rightangled triangle = w = 2

Thus, the base of the rightangled triangle = √(w^2 - x^2) = √(4 - 1) = √3

Thus, z = 1 + √3. Sufficient.

The correct answer: C

Hope this helps!

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jjjinapinch wrote:

In the figure above, what is the value of z?
1) x = y = 1
2) w = 2

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