# Every possible value of N (rectangle exercise)

If you need to:

Find every value of number N, where N is the number of right triangles that divide rectangle.

What are steps to find that number N ?

• What does "to divide a rectangle by a right triangle" mean? – Siddharth Bhat Mar 31 '19 at 12:28
• Well if the N is 2, you can divide rectangle into 2 90-degrees triangles. – user10420480 Mar 31 '19 at 12:31
• Well, in that case, $N$ can be any positive integer – Mostafa Ayaz Mar 31 '19 at 12:32
• I can use two 90-degree triangles to form a rectangle, and I can divide a given rectangle into any number of smaller rectangles, so I don't understand the question, I'm afraid – Siddharth Bhat Mar 31 '19 at 12:32
• That's the same thing I was thinking of, but I don't know am I correct so I asked it here – user10420480 Mar 31 '19 at 12:33

What does this tell you?

You can take a rectangle and divide it into an even number of right triangles following the strategy described in the image (i.e. dividing every rectangle in smaller rectangles and these in right triangles with the diagonal).

Now, how to obtain an odd number of right triangles? Easy: Take one of the right triangles and divide it into two further right triangles with the altitude. Hence $$N$$ can take any value in $$\mathbb Z^+\setminus\{1\}$$

Here's how you can achieve an odd number of triangles

• Is N any number in Z+ or any even number in Z+? – user10420480 Mar 31 '19 at 13:43
• $N$ can be odd or even – Dr. Mathva Mar 31 '19 at 13:44
• Is N on the third picture (green one) equal to 3 or 6 ?(We have 6 triangles on third picture) – user10420480 Mar 31 '19 at 13:49
• In the third picture $N=6$ – Dr. Mathva Mar 31 '19 at 13:53
• Can N be equal to 3 ? – user10420480 Mar 31 '19 at 13:57