How many isosceles triangles with integer sides are possible such that the sum of two of the sides is 16 cm?

24

We know that,

In a triangle,

Sum of two sides is always greater than the third side

Difference of the two sides is always less than the third side

Let the sides of the triangle = a cm, a cm and b cm.

Case 1) when a + a = 16 

So,  |a - a| <  b  < a + a 

= 0 < b < 2a 

= 0 < b < 16 

So b can 1, 2, 3,………,15 

In this case total 15 triangles are possible

Case 2, when a + b = 16 (x and y are not equal to 8)

|a - a| < b < a + a

= 0 < b < 2a 

= 0 < b < 2(16 - b) 

= 0 < b < 32 – 2b 

= 0 < 3b < 32 

= 0 < b < 10.33 

b can take values from 1 to 10 except 8, so here y can take 9 values

Total number of triangles = 15 + 9 = 24