Triangles within a TriangleDate: 11/10/96 at 07:38:57 From: Cato H. Jensen Subject: Geometry, Math Project (GCSE) 1996 This is an English project called "Triangles in Triangles" for I and H level. We are doing this in 10th grade. The question is: Investigate equilateral triangles of other sizes in order to examine the relationship between the size of a triangle and the total number of triangles it contains. The numbers (where n = total number of triangles and b = number of triangles in the base) given are : b n 1 1 2 5 3 13 4 27 5 48 6 78 7 118 8 170 I need to find a formula for the total number of triangles when there are X base triangles. I have done the rest of the project, but this part has gotten me really stuck. Date: 01/04/97 at 15:58:04 From: Doctor Chaos Subject: Re: Geometry, Math Project (GCSE) 1996 I think the best way to approach your problem is to examine the pattern you were given in order to calculate the total number of triangles. Look at these pictures: This triangle has 3 little triangles in the base so b=3. /\ In this triangle, there are nine (9) little /__\ triangles or b^2. Then there are three (3) /\ /\ triangles which are larger (containing 4 little ones) /__\/__\ and one (1) largest (containing all 9). /\ /\ /\ /__\/__\/__\ 9+3+1 = 13 triangles. This triangle has 4 little triangles in the base so b = 4. /\ The total of little triangles is b^2 or 16. /__\ The number of bigger triangles (containing 4 little /\ /\ ones) is 7. The number of still bigger triangles /__\/__\ (containing 9 little ones) is 3. The number of /\ /\ /\ largest triangles contains all 16 and is 1. /__\/__\/__\ /\ /\ /\ /\ 16+7+3+1 = 27 triangles /__\/__\/__\/__\ If we examine the numbers of triangles at every level, we may see a pattern which may then suggest a formula. b = Number of triangles in the base l = Number of little triangles f = Next largest triangle (contains 4) n = Next largest triangle (contains 9) s = Next largest triangle (contains 16) T = Total number of triangles b l f n s T 1 1 0 0 0 1 2 4 1 0 0 5 3 9 3 1 0 13 4 16 7 3 1 27 5 25 ? ? ? 48 6 36 ? ? ? 78 7 49 ? ? ? 118 8 64 ? ? ? 170 Organize the data you have already collected and see if there is a formula that will help you add up the various sizes of triangles to arrive at the total. You might even need 2 different formulas; one for an odd number on the base and one for even numbers. You never know until you try. Good luck. -Doctor Chaos, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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