Question

Terry is drawing isosceles triangles with side lengths a, b, and c such that

a = y − x
b = x + z
c = y − z
Where x, y, and z are positive integers and x + y + z < 10.
Find all the possible triples (a, b, c) that satisfy this.

Answer 1

9514 1404 393

Answer:

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

Explanation:

The requirement that side lengths be greater than 0 means that ...

y > x > 0

y > z > 0

Then possible values for x and z are 1, 2, 3, and possible values for y are 3, 4, 5, 6, 7.

When we list the possibilities, and exclude those that do not give isosceles triangles, we find the values of {a, b, c} can be those listed above.