Octagon

[mrsultanmehmood]

The figure above is a regular octagon. A diagonal of an octagon is any line segment connecting non adjacent sides.

Col A number of diagonals that are parallel to sides diagonals

Col B number of diagonals which are not parallel to the sides of octagon.

[tommywallach]

Hey Mr. Sultan,

I feel that once again you've copied this wrong, if only because the definition of a diagonal is patently wrong, and that alone would screw you up. A diagonal connects vertices, not sides. There are infinite number of lines that connect sides.

The best way to solve this is simply to draw an octagon and connect one vertex to the other five vertices it could be connected to (only 5 because it can't connect to itself, and it's already connected by the outline of the octagon to the points immediately to the right and left). You only need to test ONE vertex like this, because all the others will be exactly the same. When you do this, you'll see that THREE of the lines you draw are parallel to a side, while two of them are not.

On the GRE, it's impossible to PROVE the answer to a question like this, which would require real advanced geometry. Just draw the thing and start throwing out lines.

-t

[dddannie6]

Hi t,

I had a follow up question to this.

SO since there are 3 diagonals that would make a parallel line with a side of the octagon, does that mean that there are 12 diagonals that fit this criteria. I figure this since for every vertex there is 3 diagonals so that means there are 24 ways of doing this. However, I figure I double counted some, so I divided the number by 2 to get 12. Am I right in doing this?

Since there are 40 different lines I could make with this figure (5 ways of making lines per vertex and there are 8 different vertices) I then subtract 40-12 to get 28 that do NOT make parallel lines with any of the sides.

Is that reasoning correct or is there another reason why the answer is B?

Best,

Dannialles D

[tommywallach]

Hey DDD,

I've still not seen the correct text for this, have I? That original text can't be right, can it?

Again, I'd just draw it and work out the duplicates from that. I wouldn't do it theoretically. So if you got that answer by drawing, I'm sure it's correct. : )

-t

P.S. If there's more I can do to help, please copy down the exact text from the question if you would. Or is that really the text?

[dddannie6]

The original question goes as follow:
(regular octagon is shown)
The figure Above is a regular octagon. A diagonal of an octagon is any line segment connecting two nonadjacent vertices.

Quantity A: The number of diagonals of the octagon that are parallel to at least one side of the octagon.

Quantity B: The number of diagonals of the octagon that are NOT parallel to any side of the octagon.

So you are suggesting I draw out the figure, draw the diagonals from each vertex and then eliminate any diagonals. Wouldn't that take up a lot of time?

Best,
Dannialles

[tommywallach]

Hey Dannialles,

I don't mean you need to draw out each line, but that by drawing one set of 5 (each vertex connects to 5 others, because you ignore itself and the two that it is next to), you could immediately know what was going to happen.

-t

[irs031]

We draw an octagon and pick a random vertex (lets call it vertex A). Lets name the rest of the vertices B, C, D, E, F, G and H (clockwise from A). We draw a diagonal from A to all other nonadjacent vertices (C, D, E, F and G). We see that only 2 out of the 5 diagonals drawn are parallel to another side of the octagon.

A) 8 vertices x 2 parallel diagonals = 16 diagonals total
B) 8 vertices x 3 non-parallel diagonals = 24 diagonals total

B is the answer.

[tommywallach]

Exactly!

-t