If a triangle has a perimeter of 52 units, then all its sides have length ______ units.

The perimeter of the triangle is 52 units.

In any triangle, the sum of the lengths of any two sides must be greater than the third side. This is known as the triangle inequality theorem:

​$$a + b > c, \quad b + c > a, \quad a + c > b$$​​

To determine the maximum possible value of any side, we assume that one side of the triangle is as long as possible while the other two sides are as short as possible (while still satisfying the triangle inequality).

Let the sides of the triangle be a, b, and c.

The perimeter of the triangle is the sum of the lengths of the sides: a + b + c = 52

Let’s assume that one of the sides is ‘a’ and it has the maximum possible value. Using the triangle inequality:

​$$a < b + c$$​​

Since b + c = 52 - a,

substituting:

​$$a < 52−a\implies 2a<52$$​​

Thus, the maximum possible length of any side is less than 26 units.