Final answer:
The smallest possible perimeter of a triangle with side lengths as squares of distinct positive integers is 14.
Step-by-step explanation:
To find the smallest possible perimeter of a triangle whose side lengths are all squares of distinct positive integers, we need to consider the properties of squares. The square of any positive integer can be expressed as a^2. Since the triangle has three distinct side lengths, we can choose three different positive integers and find their squares. Let's consider the squares of the integers 1, 2, and 3. The squares of 1, 2, and 3 are 1^2 = 1, 2^2 = 4, and 3^2 = 9.
Next, we need to find the possible combinations of these squares to form a triangle. We have three choices: 1, 4, 9 or 1, 9, 4 or 4, 1, 9. Using the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side, we can determine the valid combinations:
1 + 4 > 9: Invalid
1 + 9 > 4: Valid
4 + 9 > 1: Valid
So, the valid combination is the side lengths 1, 9, and 4. To find the perimeter, we add the three side lengths together: 1 + 9 + 4 = 14. Therefore, the smallest possible perimeter of the triangle is 14.