A triangle has two sides of lengths 6 and 9. What value could the length of the third side be? Check all that apply.A.4B.7C.2D.10E.15F.12

In a triangle, the length of any side must be less than the sum of the lengths of the other two sides, and more than the absolute difference of the lengths of the other two sides. This is known as the triangle inequality theorem.

Given that two sides of the triangle are 6 and 9:

1. The sum of these two sides is 6 + 9 = 15.
2. The absolute difference of these two sides is |6 - 9| = 3.

So, the length of the third side must be more than 3 and less than 15.

Checking the options:

A. 4: This is more than 3 and less than 15, so it could be the length of the third side.

B. 7: This is more than 3 and less than 15, so it could be the length of the third side.

C. 2: This is less than 3, so it could not be the length of the third side.

D. 10: This is more than 3 and less than 15, so it could be the length of the third side.

E. 15: This is equal to 15, so it could not be the length of the third side. The third side must be less than, not equal to, the sum of the other two sides.

F. 12: This is more than 3 and less than 15, so it could be the length of the third side.

So, the possible lengths for the third side are 4, 7, 10, and 12.