Date: 01/07/98 at 03:21:18 From: Arash Sadeghi Subject: Algebra A basketball court is made up of square parquet tiles, all the same size, laid side by side to form a rectangle 105 tiles wide and 135 tiles long. If a straight line is drawn diagonally from one corner of the floor to the opposite corner, how many tiles will the diagonal intersect? Thanks.
Date: 01/07/98 at 06:21:50 From: Doctor Mitteldorf Subject: Re: Algebra Dear Arash, Let's divide the court into 15 lengthwise and 15 widthwise, so that each section is 7 by 9 tiles. If you draw a diagonal across the whole thing, it cuts exactly through the corner of each 7x9 rectangle, so it goes through 15 corners like that. I don't know if you want to call this going through two tiles or going through none, when the line passes exactly through a corner. For now, I will call it not going through any tiles at all. Now, within each 7x9 rectangle, the line must pass 8 sides to go the length and 6 sides to go the width. Each time it goes through a side, it enters another tile. Counting the tile it started in as 1, it must pass through 1+8+6 = 15 tiles to get across the 7x9 rectangle. Since there are 15 rectangles, and the line goes through 15 tiles to pass through each rectangle, the answer is 15*15 = 225 tiles that the line passes through. (There are another 15 pairs of tiles that it "touches" as it passes by the corner only.) -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 01/15/98 at 23:02:55 From: Doctor Rob Subject: Re: Algebra Since 105 and 135 have the common divisor 15, the diagonal will cross 15 times as many tiles as the diagonal of a 105/15 = 7 by 135/15 = 9 tile rectangle. Now since 7 and 9 have no common factor, the answer for the small rectangle is 7 + 9 - 1 = 15. The final answer is then 15*15 = 225. I leave it to you to prove that if x and y have no common factor, then the diagonal of an x-by-y rectangle intersects x + y - 1 tiles. HINT: Start at one corner with one square. What happens every time the diagonal crosses either a horizontal or vertical line? Figure out why we need x and y to have no common factor. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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