Select ALL the sets of three side lengths that will make a triangle. 3, 4, 8
7, 6, 12
5, 11, 13
4, 6, 12
4, 6, 10​

2 Answers:B) 7, 6, 12C) 5, 11, 13

Explanation:

We'll use the triangle inequality theorem. Recall that this theorem says that a triangle is only possible if and only if adding any two sides leads to a sum larger than the third side.

Let's say we had a triangle with sides a,b,c. Such a triangle is only possible if all three conditions hold true

a+b > cb+c > aa+c > b

Let's go through each answer choice to see if those conditions hold up.

Choice A. We have a = 3, b = 4, c = 8. Note that a+b = 3+4 = 7 which is not greater than c = 8. Therefore a+b > c is false, and we cannot make a triangle here. We don't need to check the other inequalities. I recommend you trying it out by cutting out slips of paper of these lengths. You'll find a triangle cannot be constructed. We can rule out choice A.Choice B. We have a = 7, b = 6 and c = 12. We can see that a+b = 7+6 = 13 which is larger than c = 12. So far, so good. Then note that b+c = 6+12 = 18 which is larger than a = 7. Finally, a+c = 7+12 = 19 which is larger than b = 6. Picking any two sides, adding them up, leads to a sum larger than the third side. A triangle is possible in this case. Choice B is one of the answers.Choice C. This time a = 5, b = 11, c = 13. You should find that a triangle is possible following the same steps as done in choice B above. Adding any two sides leads to a sum larger than the third side. Choice C is one of the answers. Choice D. Now we have a = 4, b = 6, c = 12. Then note a+b = 4+6 = 10 which is too small compared to c = 12. Meaning a+b > c is false. A triangle isn't possible. Cross choice D off the list.Choice E. In this case, a = 4, b = 6 and c = 10. Adding the first two values gets us a+b = 4+6 = 10 which is not larger than c = 10. If you used slips of paper, you'd find that a straight line is formed rather than a triangle. Cross choice E off the list.

To wrap things up, only choice B and choice C are possible triangles because adding any two sides leads to a sum larger than the third side.

Look at this and see if it