Which of the following sets of numbers could represent the three sides of a triangle? $\newline$ $\{12,20,34\}$ $\newline$ $\{4,6,11\}$ $\newline$ $\{10,12,21\}$ $\newline$ $\{13,18,33\}$

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Equivalent ratios: word problems

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Q. Which of the following sets of numbers could represent the three sides of a triangle?

\newline

\{12,20,34\}

\newline

\{4,6,11\}

\newline

\{10,12,21\}

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\{13,18,33\}

Recall Triangle Inequality Theorem: Recall the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Apply Theorem to Set $\{12, 20, 34\}$ : Apply the Triangle Inequality Theorem to the first set of numbers $\{12, 20, 34\}$ . Check if the sum of any two numbers is greater than the third number. $12 + 20 = 32$ , which is not greater than $34$ .
Set $\{12, 20, 34\}$ Analysis: Since $32$ is not greater than $34$ , the set $\{12, 20, 34\}$ does not satisfy the Triangle Inequality Theorem and cannot represent the sides of a triangle.
Apply Theorem to Set $\{4, 6, 11\}$ : Apply the Triangle Inequality Theorem to the second set of numbers $\{4, 6, 11\}$ . Check if the sum of any two numbers is greater than the third number. $4 + 6 = 10$ , which is not greater than $11$ .
Set $\{4, 6, 11\}$ Analysis: Since $10$ is not greater than $11$ , the set $\{4, 6, 11\}$ does not satisfy the Triangle Inequality Theorem and cannot represent the sides of a triangle.
Apply Theorem to Set $\{10, 12, 21\}$ : Apply the Triangle Inequality Theorem to the third set of numbers $\{10, 12, 21\}$ . Check if the sum of any two numbers is greater than the third number. $\newline$ $10 + 12 = 22$ , which is greater than $21$ .
Set $\{10, 12, 21\}$ Analysis: Check the other two combinations for the set $\{10, 12, 21\}$ to ensure all conditions of the Triangle Inequality Theorem are met. $\newline$ $10 + 21 = 31$ , which is greater than $12$ . $\newline$ $12 + 21 = 33$ , which is greater than $10$ .
Apply Theorem to Set $\{13, 18, 33\}$ : Since all combinations of the set $\{10, 12, 21\}$ satisfy the Triangle Inequality Theorem, this set can represent the sides of a triangle.
Set $\{13, 18, 33\}$ Analysis: Apply the Triangle Inequality Theorem to the fourth set of numbers $\{13, 18, 33\}$ . Check if the sum of any two numbers is greater than the third number. $13 + 18 = 31$ , which is not greater than $33$ .
Set $\{13, 18, 33\}$ Analysis: Apply the Triangle Inequality Theorem to the fourth set of numbers $\{13, 18, 33\}$ . Check if the sum of any two numbers is greater than the third number. $13 + 18 = 31$ , which is not greater than $33$ .Since $31$ is not greater than $33$ , the set $\{13, 18, 33\}$ does not satisfy the Triangle Inequality Theorem and cannot represent the sides of a triangle.