Which of the following sets of numbers could represent the three sides of a triangle? $\newline$ $\{12,20,34\}$ $\newline$ $\{13,25,38\}$ $\newline$ $\{6,13,19\}$ $\newline$ $\{15,24,36\}$

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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

\{13,25,38\}

\newline

\{6,13,19\}

\newline

\{15,24,36\}

Understand Triangle Inequality Theorem: Identify 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.
Test First Set of Numbers: Test the first set of numbers $\{12, 20, 34\}$ to see if they satisfy the Triangle Inequality Theorem. $\newline$ Check if $12 + 20 > 34$ , $12 + 34 > 20$ , and $20 + 34 > 12$ . $\newline$ $12 + 20 = 32$ , which is not greater than $34$ .
First Set Does Not Satisfy Theorem: Since $12 + 20$ is not greater than $34$ , the first set of numbers \{ $12, 20, 34$ \} does not satisfy the Triangle Inequality Theorem and cannot represent the sides of a triangle.
Test Second Set of Numbers: Test the second set of numbers ${13, 25, 38}$ to see if they satisfy the Triangle Inequality Theorem. $\newline$ Check if $13 + 25 > 38$ , $13 + 38 > 25$ , and $25 + 38 > 13$ . $\newline$ $13 + 25 = 38$ , which is not greater than $38$ .
Second Set Does Not Satisfy Theorem: Since $13 + 25$ is not greater than $38$ , the second set of numbers \{ $13, 25, 38$ \} does not satisfy the Triangle Inequality Theorem and cannot represent the sides of a triangle.
Test Third Set of Numbers: Test the third set of numbers $\{6, 13, 19\}$ to see if they satisfy the Triangle Inequality Theorem. $\newline$ Check if $6 + 13 > 19$ , $6 + 19 > 13$ , and $13 + 19 > 6$ . $\newline$ $6 + 13 = 19$ , which is not greater than $19$ .
Third Set Does Not Satisfy Theorem: Since $6 + 13$ is not greater than $19$ , the third set of numbers \{ $6, 13, 19$ \} does not satisfy the Triangle Inequality Theorem and cannot represent the sides of a triangle.
Test Fourth Set of Numbers: Test the fourth set of numbers $\{15, 24, 36\}$ to see if they satisfy the Triangle Inequality Theorem. $\newline$ Check if $15 + 24 > 36$ , $15 + 36 > 24$ , and $24 + 36 > 15$ . $\newline$ $15 + 24 = 39$ , which is greater than $36$ . $\newline$ $15 + 36 = 51$ , which is greater than $24$ . $\newline$ $24 + 36 = 60$ , which is greater than $15$ .
Fourth Set Satisfies Theorem: Since all the inequalities are satisfied for the fourth set of numbers $\{15, 24, 36\}$ , this set does satisfy the Triangle Inequality Theorem and can represent the sides of a triangle.