Shelbie has been asked to construct as many unique triangles as she can that have all sides of integer length, one side of length 5 cm, and a perimeter less than or equal to 16 cm.
shelbie can construct 11131517 unique triangles of which 6789 are strictly isosceles and 0 are1 is2 are3 are equilateral.

perimeter must lie between 7 and 16 inclusive.

p=7

{5, 1, 1}

p = 8

{5, 1, 2}

p = 9

{5, 1, 3}, {5, 2, 2}

p = 10

{5, 1, 4}, {5, 2, 3}

p = 11

{5, 1, 5}, {5, 2, 6}, {5, 3, 3}, {5, 4, 2}

p = 12

{5, 1, 6}, {5, 2, 5}, {5, 3, 4}

p = 13

{5, 1, 7}, {5, 2, 6}, {5, 3, 5}, {5, 4, 4}

p = 14

{5, 1, 8}, {5, 2, 7}, {5, 3, 6}, {5, 4, 5}

p = 15

{5, 1, 9}, {5, 2, 8}, {5, 3, 7}, {5, 4, 6}, {5, 5, 5}

p = 16

{5, 1, 10}, {5, 2, 9}, {5, 3, 8}, {5, 4, 7}, {5, 5, 6}

there are 31 possible solutions. now eliminate all solutions which do not satisfy condition  : {5, 1, 1}, {5, 1, 2}, {5, 1, 3}, {5, 2, 2}, {5, 1, 4}, {5, 2, 3}, {5, 1, 6}, {5, 1, 7}, {5, 1, 8}, {5, 2, 7}, {5, 1, 9}, {5, 2, 8}, {5, 1, 10}, {5, 2, 9}, {5, 3, 8}

the total number of triangles is 31 - 15 = 16.

there is 1 equilateral triangle.