D10 OG 12 Page 21 Geometry

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D10 OG 12 Page 21 Geometry

by EMAN » Sun Oct 04, 2009 3:26 pm
In the figure shown, what is the value of v + x + y + x + w?


(In the book there is a diagram of an uneven five pointed star with the bottom of the points having lines close them off. In other words, if you drew a star by hand with all the crossing lines, that's essentially what it looks like. Starting at 12:00 and going clockwise, it goes y, z, w, v, and x in degrees at each triangular point).

The OG explanation makes sense but it's very tedious in that you have to first use the formula (5-2) x 180 = 540. Then you have to find five separate triangles in the diagram that add up to the answer. With all this information, there's some substitution and factoring to get to the answer. There has to be an easier way to approach this problem. Please advise.

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by okigbo » Sun Oct 04, 2009 5:15 pm
The 5 triangles add up to 540 degrees as you mentioned. Each triangle is 108 degrees. Each angle is 36 degrees and the five angles add up to 180 degrees.

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by EMAN » Sun Oct 04, 2009 6:20 pm
Can you explain you derive how each triangle is 108 degrees please?

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by okigbo » Sun Oct 04, 2009 6:29 pm
540 degrees divided by 5 triangles.

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by EMAN » Sun Oct 04, 2009 6:43 pm
I think that happens to coincidentally work out. Are you sure you can do solve it like this? I would think each of the five triangles needs to be add up to 180 degrees, not 108. Additionally, they weren't the same proportions in the diagram so we can't just assume they are equilateral triangles. Let me know what you think.

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by benjiboo » Sun Sep 02, 2012 3:19 pm
For those of you having trouble with this one... Just imagine trying to solve an easier problem... What if you were given the attached problem... Seems easier eh? The OG higher level question just test to see if you can think further out...

In this problem:
Quesiton D10 concept.jpg
Question D10.jpg


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by Ian Stewart » Sun Sep 02, 2012 11:14 pm
There's a useful GMAT-specific logical principle that can be applied here. Notice that this question has 5 exact numerical answer choices; we don't have an answer which says 'cannot be determined'. Well, a GMAT question can't have more than one right answer. So if one of those five answers is right, it must be right no matter how we choose to draw the diagram. Whether the diagram is drawn in some skew way, or completely symmetrically, the answer must always be the same, or they couldn't ask the question this way to begin with. If the answer were different depending how you drew the picture, they'd need an answer choice which said "cannot be determined".

So we're free to just draw the most symmetric diagram possible and solve the question using that. So we can make the middle pentagon a regular pentagon, so each angle is 540/5 = 108 degrees. Then the exterior triangles become isosceles, with angles 72-72-36, where the angles at each point of the star are 36 degrees. Since we need to sum the angles at the five points, the answer is 5*36 = 180.

It's a much longer problem if one answer choice does say "cannot be determined", since then you actually have to prove the answer is the same even when the diagram is drawn in some irregular way.
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by GMATGuruNY » Mon Sep 03, 2012 5:12 pm
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Hi All,

To start, it’s incredibly rare to see a Quant question on the GMAT with so many inter-connected shapes (including a hexagon!), so you shouldn’t worry about this question if you got it wrong the first time you attempted it. You can solve it with a mix of Geometry rules and TESTing VALUES.

With geometric shapes, every time you “add a side”, you increase the total number of degrees by 180. For example….

A triangle = 180 degrees
A square/rectangle = 360 degrees
A hexagon (5-sided shape) = 540 degrees

Since we are not given any information about any of the angles in this picture, we can TEST VALUES. Since the hexagon includes 5 angles that total 540 degrees, it’s easiest to make all 5 angles the same…. 540/5 = 108 degrees each.

The sum of the angles on a line add up to 180 degrees, so each angle in each of the triangles that is next to a 108 degree hexagon angle is equal to 72.

That means that each triangle is an isosceles triangle with two 72 degree angles and one angle that equals… 180 – 72 – 72 = 36 degrees. Thus, V = W = X = Y = Z = 36 and the sum of those 5 angles is (5)(36) = 180 degrees.

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