If the triangle above has side lengths w, x and y, what is the area of

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 technique can save a lot of time and is discussed in greater detail in the video below.

Target question: What is the area of triangle ABC?

Statement 1: \(w = 2\)
First recognize that, since all 3 angles are given, the shape of the triangle is permanently set, but the SIZE of the triangle is not set.
That is, we can take the given triangle and make it as small or as large as we wish, and the 3 angles will stay the same (30°, 45° and 105°)

So, as you might imagine, there are infinitely-many different sized triangles with angles 30°, 45° and 105°.

IMPORTANT: Each of those different triangles has a UNIQUE set of measurements.
For example, there's EXACTLY ONE 30-45-105 triangle in which w = 1, and there's EXACTLY ONE 30-45-105 triangle in which w = 3.574, and there's EXACTLY ONE 30-45-105 triangle in which w = 9.300741, and so on.
This also means that there's EXACTLY ONE 30-45-105 triangle in which w = 2.
This means statement 1 (\(w = 2\)) locks in the shape and size of triangle ABC.
As such statement 1 is SUFFICIENT.

Aside: Do we need to calculate the area of triangle ABC?
Absolutely not! We need only recognize that there's ONLY ONE 30-45-105 triangle in the universe such that w = 2, and that we COULD find its area.

Statement 2: \(x + y = 3\sqrt{2} + \sqrt{6}\)
Once again, we need to recognize that there are infinitely-many different sized triangles with angles 30°, 45° and 105°.
Each one of these individual triangles will have UNIQUE measurements for w, x, and y.
This also means that each triangle must have a UNIQUE SUM of x and y.
This means statement 2 (\(x + y = 3\sqrt{2} + \sqrt{6}\)) locks in the shape and size of triangle ABC.
As such statement 2 is SUFFICIENT.

Answer: D

For more practice with this technique, try answering the following questions:
550-level: https://gmatclub.com/forum/the-figure-a ... l#p2273208
650-level: https://gmatclub.com/forum/in-the-figur ... l#p1933494
700-level: https://gmatclub.com/forum/if-the-area- ... l#p1741574
700-750 level: https://gmatclub.com/forum/for-triangle ... l#p1733119

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