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### Counting Rectangles Cut By a Diagonal

Date: 06/15/99 at 13:27:47
From: Ben
Subject: Rectangles Cut By a Diagonal Line

I am writing to ask you if you can help me on this project we have
been given. What we are asked to do is find an equation for the number
of 1 cm squares that are cut by a line going through the middle (from
corner to corner.)

For example, with a 3 x 5 rectangle (each square = 1 cm) the number of
rectangles cut is 7.

Please can you try to help us? Thanks!

Date: 06/15/99 at 17:38:05
From: Doctor Anthony
Subject: Re: Rectangles Cut By a Diagonal Line

The general rule: If the lengths of sides (a x b) of the rectangle are
mutually prime, the number of squares cut is a+b-1

Thus, your example: (3 x 5) gives 3+5-1 = 7

Other examples:     (8 x 5) gives 8+5-1 = 12
(9 x 4) gives 9+4-1 = 12
(9 x 5) gives 9+5-1 = 13

BUT (9 x 6) DOES NOT give 9+6-1 = 14. Instead you must proceed as
follows:

First divide (9 x 6) through by common factor 3 to get (3 x 2)
Then apply the rule to (3 x 2) to give 3+2-1 = 4
Now multiply by the factor 3 again to get 12 (which is correct).

Let's do a square figure, say (5 x 5). We divide through by 5. This
gives (1 x 1). Applying the rule gives 1+1-1 = 1. Now multiply up by
the factor 5 again to get 5. We know this is correct because in ANY
square figure the number of squares that are cut will be equal to the
side of the square.

The general procedure for a rectangle (a x b) is as follows:

If a and b are relatively prime the answer is a+b-1

If a and b have a common factor c, first divide through by c to get
(a/c x b/c).
Then apply the rule to get  a/c + b/c - 1.
Finally, multiply through again by c to get (a+b-c).

For example, with (9 x 6) the correct answer is 9+6-3 = 12.

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/

Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Puzzles
High School Triangles and Other Polygons
Middle School Geometry
Middle School Puzzles
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

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