[Guest]

This is a square with lines drawn from every corner to the mid point of each side.

What is the ratio of the shaded area to the whole?

P.S.1: Sorry if this has been done b4

P.S.2: Math gurus... this one is easy, keep you answer to yourself for a lil while.

[Guest]

My answer: 1/3

[Guest]

1/3

imagine cutting the square in 3 parts having the 3 line cut the square vertically.. then on the 2 sides the shaded part will form a triangle. cut those in half and put in along the middle shaded part and it will complete a third of the square

[Guest]

I came up with 30.666% shaded area.

[Guest]

I came up with 26.666% shaded area. I made an image that is 300x300 (90,000) pixels, turns out the inner red square is 120*120(14,400). The extra blue triangles make a rectangle 120*80(9,600)

(14,400+9,600) / 90,000 = .2666666666 or 26.66% I hope this is right

Sorry for double post and wrong first guess (bad at math) wouldn't let me edit 1st post

Edited June 8, 2009 by kkehoe5

[Guest]

Sorry for double post and wrong first guess (bad at math) wouldn't let me edit 1st post

I came up with 26.666% shaded area. I made an image that is 300x300 (90,000) pixels, turns out the inner red square is 120*120(14,400). The extra blue triangles make a rectangle 120*80(9,600)

(14,400+9,600) / 90,000 = .2666666666 or 26.66% I hope this is right

Your drawing is correct but red area would be 1/5 of the square, leading to the result as 1/3.

[k-man]

Feegs, please use the spoiler next time you post the answer.

The correct answer is 1/3 and here is how it works out:

The figure is very symmetrical, so to simplify we can look at the smallest repeating part, which is shown as a red triangle on the image below. This triangle makes up 1/8 of the whole figure. The ratio of the dark red portion of this triangle relative to the whole triangle is the object of the puzzle.

To make the calculations easier to explain I made another drawing of this triangle without the extra noise, but with one helpful addition. On the picture below the line CE divides AD in 2 equal parts AE and ED. FG is perpendicular to DC. We are looking for the ratio of the area of the shaded triangle DFC to the area of the triangle DBC.

1) AD = 2 * ED

2) From 1) and the the fact that triangles EDC and FGC are similar GC = 2 * FG.

3) From BC=DC and the fact that triangles DFG and DBC are similar DG = FG.

4) From 2) and 3) DC = 3 * DG

5) Naming DG length as a the area of the triangle DFG is a²/2

6) From 2) the area of the triangle FGC is (2a * a) / 2 or simply a²

7) From 5) and 6) the total area of the shaded triangle DFC is 3/2 * a²

8) From 4) the area of the triangle DBC is (3 * a)²/2 or 9/2 * a²

9) From 7) and 8) we get the answer - the area of DBC is three times the area of DFC.

Edit: BMP images didn't show up, so I replaced them with PNGs.

Edited June 8, 2009 by k-man

[Guest]

This is a square with lines drawn from every corner to the mid point of each side.

What is the ratio of the shaded area to the whole?

P.S.1: Sorry if this has been done b4

P.S.2: Math gurus... this one is easy, keep you answer to yourself for a lil while.

is it 50%?

[Guest]

Quote from k-man: Feegs, please use the spoiler next time you post the answer.

I tried inserting the spoiler it several ways but it wasn't inserting. Got any hints for me.

Elegant proof to the solution by the way, K-man

Edited June 8, 2009 by Feegs

[k-man]

I tried inserting the spoiler it several ways but it wasn't inserting. Got any hints for me.

I don't know what wasn't working for you, but if all else fails you can always just type the following tags around the solution/answer:

[spoiler="hint"]this is a hidden text including hints and solution[/spoiler]

It will appear like this:

this is a hidden text including hints and solution

Elegant proof to the solution by the way, K-man

Care to share your solution? You were the first to provide the correct answer, but the solution is usually what is sought.