I don't understand how to solve this

Gabriela, have you taken a look at the Solution from the question itself? Here it is repeated for convenience:

"Each side of a triangle is shorter than the sum of the lengths of the other two sides, and, at the same time, longer than the difference of the two. Hence, the length of the third side of the triangle in the question is greater than the difference of the other two sides (12 – 4 = 8) and smaller than their sum (12 + 4 = 16). Since only choices (C) and (D) lie between the values 8 and 16, the answers are (C) and (D)."

When does pathagorean theorem not solve for trianges?

Julie, Pythagorean theorem only applies to right triangles. In this problem we are not affirmatively told that the triangle is a right one (one of the angles measures 90 degrees).

To answer this question, you go back to the property of a triangle. Imagine a clock with two hands: 4 and 12 in length each. The two hands form the two sides of a triangle. What would be the minimum and maximum lengths of the third side?

In one extreme case, it's close to 12 o'clock. The third side measures slightly higher than the difference between the two hands, ie, 8. In another extreme case, it's close to 6 o'clock. The third side measure slightly lower than the sum between the two sides, ie, 16.

I don't understand, so there isn't a formula to solve this?

I tried to use an actual ruler to solve the problem. Even so, it is hard to figure out the value of the other side of the triangle. I hesitate guessing an answer. Any standard formula for this?

Sinai, LJ, there is no formula. You only need to remember the properties of a triangle. It was explained in a prior post in this thread:

Imagine a clock with two hands: 4 and 12 in length each. The two hands form the two sides of a triangle. What would be the minimum and maximum lengths of the third side?

Case 1: it's close to 12 o'clock. The third side measures slightly longer than the difference between the two hands, ie, 8. Case 2: it's close to 6 o'clock. The third side measure slightly shorter than the sum between the two sides, ie, 16.

Try drawing these cases, so you will understand.

I tried to do this problem using the Pythagorean theorem, but as the question doesn't state that the triangle is a right triangle, I'm unable to get the correct answer. Do you have to use the law of sines or cosines for this?

I like ur "clock way" to explain this principle.

But what if the new side is one of the sides that i wil add and rest to follow the rule?

I have the option to choose F as an answer, but there is no F in the problem. I'm not sure if this is just a random bug, but I thought you should know about it.