Question

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​

Answer 1

2 Answers:

• B) 7, 6, 12
• C) 5, 11, 13

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Step-by-step 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 > c
• b+c > a
• a+c > b

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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.

Answer 2

7 + 6

5 + 11

Step-by-step explanation:

the lengths of the two shorter sides of a triangle must add up to more than the length of the third side.

3 + 4 = 7 ≤ 8 no

7 + 6 = 13 > 12 yes

5 + 11 = 16 > 13 yes

4 + 6 = 10 ≤ 12 no

4 + 6 = 10 ≤ 10 no