Diagonals are the line segments that connect two non-adjacent vertices of polygons. Rectangles have two diagonals that connect two opposite vertices. They are the same size.
In this activity, we will count the number of squares the diagonal passes through.
Here is an example:
Many things can vary in these rectangles: the width, the length, and the number of squares the diagonal passes through (or is "cut"). One of the strategies we can use is keeping one of these variables constant to find the pattern. Click here to view the canvas below.
To figure out the relation among L, W, and D, you may need to draw many examples.
Notice that the total number of squares cut by the diagonal is always less than the sum of length and width. But that difference changes depending the size of the rectangles!
It might also help you to color the squares cut by the diagonal. The patterns formed by the squares can help you to figure out the pattern.
Do you notice any differences between "L + W" and "D" when the length and width have common factors and when they do not have any common factors?
Do you notice any differences between how the squares are aligned when the length and width have common factors and when they don't?
Use the canvas here to make drawings and record your results. At the start, you may not be ready to answer the first question, but see if you can build up to that.
Key for Different sized rectangles
The number of squares cut by the diagonal is d = L + W−GCF(L,W).
Now we can go back to our initial question:
In a 15 by 300 rectangle, how many squares are cut by the diagonal?
d = 15 + 300 - GCF (15, 300). The GCF of 15 and 300 is 15, so:
d = 300
In a 15 by 300 rectangle, 300 squares will be cut by the diagonal!
Extras and Resources
This property is being used in pixel coloring to create computer graphics.