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

Question

https://i.imgur.com/kGVCn5S.png

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

(1)  w = 2

(2)  x + y = 3\sqrt{2} + \sqrt{6}

Note : This post is part of my Pro Tip series. You'll find my analysis and full solution below.

BrentGMATPrepNow · Answer

No takers for this question? 
Perhaps I should have categorized the difficulty level as 700+

IanStewart · Answer

If you know the angles in a triangle, it is theoretically possible to work out the ratio of the lengths of the sides. For strange angles, you need to use trigonometry to do that; for some angles we can do it by hand (for 30-60-90 triangles, 45-45-90 triangles, and for triangles like the one above that can be divided into familiar triangles).

So from the angles, we could find the ratio of w to x to y, and if we thus know w, or know x+y, we can find all three of the numbers w, x and y, and with those three lengths we can find the triangle's area. So the answer is D.

If you genuinely did want to solve this as a PS problem, you can draw a vertical height from B to AC. That splits the triangle into a 30-60-90 triangle on the left, and a 45-45-90 triangle on the right. You can then use the 45-45-90 and 30-60-90 triangle ratios to work out every length in terms of w.

BrentGMATPrepNow · Answer

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°) https://i.imgur.com/0NLlxol.png 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 RELATED VIDEO https://www.youtube.com/watch?v=wetqSGMdY7Y

sarthaknash90 · Answer

I am still not able to understand how B option can help in resolving the questions, can someone show me step by step approach using option B

BrentGMATPrepNow · Answer

To show that statement 2 is sufficient, we use the exact same approach we used to show that statement 1 is sufficient. 
That is, each time we enlarge the original triangle by a little bit, each side length increases a little bit. 
This also means that, each time we enlarge the triangle, the perimeter increases a little bit. 
We can also say that, each time we enlarge the triangle a little bit, the sum of x and y increases a little bit.

This means, for each unique size of the triangle, there exists a unique sum of x and y. 
In other words, there is exactly one unique triangle in which the sum of x and y is 3√2 + √6
 
For this reason, statement 2 is also sufficient.

Does that help?

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

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