geometry

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x and y are whole numbers, 10<x<14, and 8<y<14.
The sides of a triangle are 5 cm, x cm, and y cm.
List possible values for x and y.

  • geometry -

    since any side is less than the sum of the other two sides,

    x+y > 5 (duh)
    x+5 > y
    y+5 > x

    lots of possible solutions:

    5,11,9
    5,13,13
    and more

  • geometry -

    values of x could be 11, 12, or 13
    values of y could be 9, 10, 11, 12 or 13

    now overthinking the problem:

    To have a triangle:
    1. x+y > 5
    2. x+5 > y ----> x-y > -5 or y-x < 5
    3. y+5 > x ----> y-x > -5

    from #2 and #2 we can see that
    -5 < y-x < 5

    so make 3 columns
    5_x__y_(y-x)

    5 11 9
    5 11 10
    5 11 11
    5 11 12
    5 11 13

    5 12 9
    5 12 10
    ..
    5 12 13

    5 13 9
    ..
    5 13 13

    in short, the x could be 11, 12 or 13
    the y could be 9, 10, 11, 12, or 13

    so the number of triangles for sides labeled x , y and 5 is 3(5) or 15
    However, triangles 5 , 12, 13 and 5, 13, 12 are the same triangle
    there are 3 such duplicated pairs, so
    the actual number of triangles possible is 15-3 = 12

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