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/
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