Question

Which of the following are possible side lengths for a triangle?

8,1,2
12, 8,5
6, 5, 11

Answer 1

Answer: 12,8,5 (choice B)

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Step-by-step explanation

The triangle inequality theorem states that a triangle of sides a,b,c is possible if and only if all of the following conditions are met:

• a+b > c
• a+c > b
• b+c > a

In other words, adding any two sides must have the sum exceed the third side.

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Let's try out the numbers in choice A

`a = 8, b = 1, c = 2\\\begin{array}\cline{1-3}a+b > c & a+c > b & b+c > a\\8+1 > 2 & 8+2 > 1 & 1+2 > 8\\9 > 2 & 10 > 1 & 3 > 8\\\text{true} & \text{true} & \text{false}\\\cline{1-3}\end{array}`

The last inequality is false, so a triangle is NOT possible here.

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Now onto choice B

`a = 12, b = 8, c = 5\\\begin{array}\cline{1-3}a+b > c & a+c > b & b+c > a\\12+8 > 5 & 12+5 > 8 & 8+5 > 12\\20 > 5 & 17 > 8 & 13 > 12\\\text{true} & \text{true} & \text{true}\\\cline{1-3}\end{array}`

All three inequalities are true when a = 12, b = 8, and c = 5.

A triangle is possible. The answer is choice B) 12, 8, 5

I recommend making slips of paper with these lengths to try it out for yourself to see if a triangle is possible or not.

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For the sake of completeness, let's look at choice C as well

`a = 6, b = 5, c = 11\\\begin{array}c\cline{1-3}a+b > c & a+c > b & b+c > a\\6+5 > 11 & 6+11 > 5 & 5+11 > 6\\11 > 11 & 17 > 5 & 16 > 6\\\text{false} & \text{true} & \text{true}\\\cline{1-3}\end{array}`

The first inequality is false, so a triangle is NOT possible.

Side note: once you find a false inequality, you do not have to check the other inequalities.