Question

If the perimeter of a triangle is 10 and lengths of the sides are natural numbers,

find all the possible lengths of sides of the triangle.

Answer 1

The possible natural number side lengths for a triangle with a perimeter of 10, satisfying the triangle inequality, are 1, 4, 5; 2, 3, 5; 2, 4, 4; and 3, 3, 4.

Step-by-step explanation:

To find all the possible lengths of the sides of a triangle with a perimeter of 10, where the side lengths are natural numbers, we have to remember that the sum of any two sides of a triangle must be greater than the third side (Triangle Inequality Theorem).

Let's denote the sides as a, b, and c, and we can assume without loss of generality that a ≤ b ≤ c. Because the perimeter is 10, we have the equation a + b + c = 10. Considering the smallest natural number is 1, we can start by setting a = 1 and finding which values of b and c satisfy the triangle inequality and the perimeter condition.

Here are the possible natural number side lengths for a triangle with a perimeter of 10:

• 1, 2, 7 (not a valid triangle as 1 + 2 is not greater than 7)
• 1, 3, 6 (not a valid triangle as 1 + 3 is not greater than 6)
• 1, 4, 5 (valid triangle)
• 2, 2, 6 (not a valid triangle as 2 + 2 is not greater than 6)
• 2, 3, 5 (valid triangle)
• 2, 4, 4 (valid triangle)
• 3, 3, 4 (valid triangle)

These combinations are all the possible natural number side lengths for a triangle with a perimeter of 10.