Question

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.

Answer 1

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.