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Rachael has 20 thin rods whose lengths, in centimetres, are 1, 2, 3, ... 20. Any two rods can be connected at their ends. Rachael selects three rods to make a triangle, then three other rods to make a second triangle, and so on. a Only one of Rachael's rods cannot be the side of a triangle. Determine which one. b Rachael uses six of her rods that are no more than 7 cm to make two triangles. In how many ways can she do this? c Show how Rachael could use 15 of her rods to make five triangles, each with an even perimeter. d Rachael wants to use 18 rods to form a set of six triangles with equal perimeters. Either find such a set or explain why no such set exists.

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sqdancefan

9514 1404 393

Answer:

  a) 1

  b) 3

  c) {{3, 6, 7}, {8, 12, 14}, {9, 10, 17}, {11, 13, 16}, {15, 18, 19}

  d) cannot do. The numbers cannot add up.

Step-by-step explanation:

a) The minimum difference between rod lengths is 1. A unit-length rod cannot be used, because the sum of that and another length cannot exceed a third length. 1 + 2 = 3; is not greater than 3.

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b) The 3 possible pairs of triangles are

  {2, 3, 4}, {5, 6, 7}

  {2, 4, 5}, {3, 6, 7}

  {2, 6, 7}, {3, 4, 5}

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c) There are numerous ways to make 5 triangles having an even perimeter. One is shown above. Several more are ...

  {5, 12, 15}, {7, 17, 18}, {8, 10, 16}, {9, 13, 20}, {11, 14, 19}

  {5, 15, 16}, {6, 9, 13}, {7, 14, 19}, {8, 11, 17}, {10, 12, 18}

  {3, 9, 10}, {5, 15, 16}, {7, 13, 18}, {8, 11, 17}, {12, 14, 20}

  {4, 9, 11}, {5, 12, 13}, {7, 18, 19}, {8, 15, 17}, {10, 16, 20}

  {5, 12, 13}, {7, 18, 19}, {8, 11, 15}, {9, 17, 20}, {10, 14, 16}

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d) No such set exists.

The sum of the 19 rod lengths is 209. Even if we throw out the rod of length 20 (to make a set of 18 rods), the perimeter must be at least 2(19)+1 = 39. Six triangles with that perimeter would require a total rod length of 6(3) = 234. The total length of rods is insufficient to make the required triangles.

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