Answer: any values between a and 3a
Explanation:
To determine the possible side lengths for the third side of the triangle, we need to use the Triangle Inequality Theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's call the length of the third side x. Then, based on the Triangle Inequality Theorem, the following two inequalities must be true:
a + 2a > x (the sum of the first and second sides is greater than the third side)
a + x > 2a (the sum of the first and third sides is greater than the second side)
2a + x > a (the sum of the second and third sides is greater than the first side)
Solving the first inequality for x gives us:
x < 3a
Solving the second inequality for x gives us:
x > a
Solving the third inequality for x gives us:
x > -a
Combining these three results, we find that the possible values for the third side length x are:
a < x < 3a
So, the possible lengths for the third side in terms of a are any values between a and 3a, inclusive.