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

Answer 1

There are 25. For an isosceles triangle with sides a, b & c, with a = c. The sides are all positive whole numbers. The perimeter = a + b + c = 2a + b. Ans 2a must be greater than b, or it won't be a closed figure. If 2a = b, it will be a just a line segment, not a triangle.

So start at 2a = b, which gives b = 49.5, and a = c = 24.75. The largest value of b that is valid will be: b = 49 [a whole number], and 2a = 99-49 = 50, so a = c = 25.

The values of b will have to be odd, since 2a will always be even, and they must add up to an odd number [99]. So the valid values of b are {1,3,5,...45,47,49}, which is 2n-1, with n = 1..25.

The sides of the triangle will be: {(1,49,49); (3,48,48); (5,47,47);...; (47,26,26); (49,25,25)}