The lengths of two sides of a triangle are 12 inches and 4 inches. which of the following dimensions is the third side of this triangle? 9 inches 8 inches 7 inches 6 inches

You don't say whether this is a right triangle or not.

Assuming it is a right triangle, then we use the Pythagorean Theorem to determine the length of the hypotenuse:

(hypo) = (length of third side) = √(12^2 + 4^2) = √(144+16) = √160 = 4√10.

This is approx. 12.65 inches. Since this does not match any of the possible answer choices, we'll have to take a different approach to answering this question.

Given that 2 sides of the given triangle are 12 and 4 inches, respectively, we see that the 3rd side has to be longer than 8 inches; otherwise we'd have three line segments on the same line, not forming a triangle.

By this reasoning, 9 inches is the only possible answer that could be correct. With sides 12, 9 and 4 inches, the triangle would be obtuse and appear quite flat, but not be part of a straight line as with a third side of 8.