Question

2. Multiple Choice Which of the following could represent the side lengths of a triangle? a. 2cm, 1cm, 1cm b. 2cm, 3cm, 4cm c. 7cm, 12cm, 3cm d. 4cm, 4cm, 10cm​

Answer 1

only 2, 1, 1 works. The others create triangles which are not closed.

Answer 2

Answer: Choice B

2 cm, 3 cm, 4 cm

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Step-by-step explanation

Consider a triangle with sides a,b,c.

A triangle is only possible when these 3 conditions are all true.

• a+b > c
• a+c > b
• b+c > a

In other words: adding any two sides must be larger than the third side.

If one or more of those inequalities are false, then a triangle is not possible.

For more info, search out "triangle inequality theorem".

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Part 1

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

The third column results in a false statement, so it's not possible to have a triangle with sides a = 2, b = 1, c = 1.

I recommend trying for yourself to form such a triangle. Use slips of paper that are 2 inches, 1 inch and 1 inch long.

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Part 2

`a = 2, b = 3, c = 4\\\begin{array}c\cline{1-3}a+b > c & a+c > b & b+c > a\\2+3 > 4 & 2+4 > 3 & 3+4 > 2\\5 > 4 & 6 > 3 & 7 > 2\\\text{true} & \text{true} & \text{true}\\\cline{1-3}\end{array}`

All three inequalities are true when a = 2, b = 3, c = 4.

Therefore, a triangle is possible here.

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Part 3

`a = 7, b = 12, c = 3\\\begin{array}\cline{1-3}a+b > c & a+c > b & b+c > a\\7+12 > 3 & 7+3 > 12 & 12+3 > 7\\19 > 3 & 10 > 12 & 15 > 7\\\text{true} & {\large \textbf{false}} & \text{true}\\\cline{1-3}\end{array}`

A triangle isn't possible here. The two sides 7 and 3 come up short compared to the side 12.

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Part 4

`a = 4, b = 4, c = 10\\\begin{array}c\cline{1-3}a+b > c & a+c > b & b+c > a\\4+4 > 10 & 4+10 > 4 & 4+10 > 4\\8 > 10 & 14 > 4 & 14 > 4\\{\large \textbf{false}} & \text{true} & \text{true}\\\cline{1-3}\end{array}`

The first inequality is false, so a triangle isn't possible when a = 4, b = 4, c = 10.

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In summary, the only triangle that can be formed is with sides 2, 3, 4. Therefore the answer is choice B only