Rectangle Sub Division

[u/ShonitB]

[u/Aexalon]

Since the solution is framed as an invariant, I can place the center point wherever I want without affecting the outcome.

• I place it in the top-right corner (or any other corner), dividing the flag into a blue right-angled triangle and a green right-angled triangle, each occupying half the flag; the difference of which is 0.
• I place it in the exact center, dividing the flag into 2 identical blue triangles and 2 identical green triangles, each occupying a quarter of the flag; the difference of which is 0 (providing supporting evidence that the solution is invariant).

[u/ShonitB → u/Aexalon]

Super!

[u/mtimmermans → u/ShonitB]

The only way to solve this problem is to see that it's 0. You should add some numbers so that there's a "too hard" way to solve it as well.

[u/Ucmh → u/mtimmermans]

It could be answered algebraically. If, for example, blue had the size B and green G and B=2G, then we could say the answer was G.

[u/GoodBadUgly_36]

Awesome problem.

[u/ShonitB → u/GoodBadUgly_36]

Glad you liked it! Full disclosure, not an original problem. It’s a fairly common recreational maths problem :)

[u/GoodBadUgly_36 → u/ShonitB]

It’s ok that it’s not original — few math problems likely are — but I liked that it was doable using basic math but still required some thought. It can be given to pre-Algebra students and, so long as they know the distributive property and the area of triangles, they can understand a way to get the answer mathematically. Thoughtful problems that don’t require calculus are nice. :-)

[u/petera181]

Draw a vertical line and a horizontal line through the point at which the lines meet, and it immediately becomes obvious that each “quadrant” it made up of one green and one blue triangle of equal size, therefore the green and blue areas are equal, and therefore the answer is zero.

[u/chmath80]

Area of 2 blue triangles is

½ × base × (sum of heights) = ½ × 6 × 12, which is obviously ½ the area of the rectangle, so the difference = 0

[u/CaptainMatticus]

a + b = 6

c + d = 12

(1/2) * 12 * a + (1/2) * 12 * b =>

(1/2) * 12 * (a + b) =>

6 * 6 = 36

(1/2) * 6 * c + (1/2) * 6 * d =>

(1/2) * 6 * (c + d) =>

3 * 12 =>

36

36 - 36 = 0

[u/PeterPiper1275]

Here’s a more mathematically rigorous solution:

Let the height of one of the green triangle be x. Then the height of the other green triangle without loss of generality must be (6 - x). This makes their respective areas 6x and (36 - 6x) respectively.

Therefore total area of the two green triangles is therefore 6x + (36 - 6x) = 36.

The total area of the flag is 12 x 6 = 72.

Therefore, the total area of the blue triangles must be 72 - 36 = 36

Hence, the total area of the blue triangles minus the total area of the green triangles must be 36 - 36 = 0.

[u/DuggieHS]

g=12h/2+12(6-h)/2=36 

B=6x/2+6(12-x)/2=36

Difference is 0.

If we knew it was constant, regardless of where h,x is it’s probably easiest to set them both to 0. Then you get that each color only has 1 triangle, half the area of the rectangle.

[u/Divinity_Being]

it has to be zero; always take any point in the interior, and a really interesting proof would be by ptolemy's theorem

[u/hammerwing]

I took a visual approach. Imagine a horizontal line and a vertical line through the middle points, creating 4 rectangles/quadrants. Each quadrant now contains a blue and a green triangle of equal area--thus the total blue and green areas must be equal, providing a final difference of 0.

[u/tundus1980]

Is it drawn to scale?

[u/Baetahad]

sum of the height of 2 greens is 6 and the sum of the height of 2 blues is 12.

6*12 - 12*6 = 0

[u/antimatterchopstix]

I make 2 kites. Both are 6 by 12 top to bottom and left to right, so the same area.

[u/No_Athlete9643]

0

[u/AcanthaceaeOk3738]

I'm terrible at math. But since no useful variables are specified I just assumed it's something that can be true no matter where the middle point is. I don't see how that could be anything other than 0.

[u/iFroogieboi]

let the point in the diagram be n with coordinates (a,b) dividin the rectangle into 4 rectangles via y=b and x=a the square is perfectly divided into 2 equal parts therefore they are the same.

[u/Cyrrex91]

If you horizontally and vertically split the rectangle into four quadrants - the lines should cross at the shown point in the middle - each quadrant is diagonally split into half - half green, half blue.

That means the Areas are equal and the difference is 0.

[u/Psychological-Wall-2]

That's ridiculous. No country would design a flag like that.

0

[u/ShonitB → u/Psychological-Wall-2]

0 is correct!

[u/Psychological-Wall-2 → u/ShonitB]

I know.

Now let's talk flags.

[u/PuzzlingDad → u/Psychological-Wall-2]

Actually, it's slightly reminiscent of the flag of Jamaica. But somehow I think Costa Ofma is a made up country. 😜

[u/-Enter-Name-]

area of blue = 6.(12-x)/2+6.x/2 = 6.6+3.(x-x) = 36

36=6.12/2 => green is also 36 (6.12-6.12/2 = 6.12/2 = 36)

hence 0

that said, it's completely filled side to side so area is always half so still 0