Can anyone tell me how many triangles are in this box. Theres at least 73 but I am not sure about that answer. I put the link to the picture in the answer part because I couldn't in the question?

Answer 1

Here's the link,. You might want to print it out and really look at it. Count every single triangle, the small ones, and the large, even though they have several different ones in them. http:/i17.tinypic.com2u89l47.jpg Put a " / " after .com for the link to work. There are two kinds of triangle:

(1) some have two diagonal edges

(2) some have just one diagonal edge (and a horizontal edge and a vertical edge)

I'm going to use a coordinate system where (0,0) is the bottom-left corner and (5,5) is top-right. (So (0,5) is top-left and (5,0) is bottom-right.)

To count the (2)-type triangles, think about the diagonal edges. For instance, the diagonal line at the bottom-right corner could be the diagonal of a triangle. How many triangles can be made using this line segment and no other diagonal line segments? How many other diagonal line segments are there in the diagram? How many triangles do you get from each one of them? It might help if you can find a quick way of answering questions like "how many line segments are there inside the big diagonal from top-right to bottom-left?". (For instance, there is the line segment from (2,2) to (4,4). How many more of these are there?)

To form a type (1) triangle, you first ask where the right angle might be, and then ask which directions the two diagonal lines go and how long they are. For example, you could have the right angle at (3,3). Then you can get the triangle going to the left of there, using the points (2,2) and (2,4). But you can't go any further - you can't get the triangle from (1,1) and (1,5) because there's no line from (3,3) to (1,5). Similarly, you get a triangle by going up instead of left, but you can only go a distance 1. So: What are the possible positions for the right angle of a triangle of type (1), and how many such triangles are there for each of these points?