How many distinct isosceles triangles exist with a perimeter of 99 inches and side lengths that are positive while numbers?

Answer 1

Did you mean positive whole numbers ?

let the base be y and each of the equal sides be x

To be a triangle, x > 2y
or
y < x/2

2x + y = 99
y = 99-2x
y-intercept is 99, x-intercept is 49.5

so the x, or the base, can only be a number from 1 to 49

but to even have a triangle, the sum of any two sides must be greater than the third side
2y > x
y > (1/2)x

solving y = (1/2)x and y = 99-2x
x/2 = 99 - 2x
x = 198 - 4x
5x = 198
x = 39.6
so x < 39

so we can use a base from 1 to 39

check:
if x = 1, y = 97 , valid triangle
if x = 2, y = 95 , valid triangle
...
if x = 39 , = 21 , valid triangle
if x = 40 , y = 19 , not a possible triangle
....

there are 39 such triangles