Which of the following sets of numbers could represent the three sides of a triangle? $\newline$ $\{5,17,23\}$ $\newline$ $\{6,20,25\}$ $\newline$ $\{5,17,24\}$ $\newline$ $\{10,13,25\}$

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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?

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\{5,17,23\}

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\{6,20,25\}

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\{5,17,24\}

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\{10,13,25\}

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 First Set: Apply the Triangle Inequality Theorem to the first set ${5,17,23}$ . Check if $5 + 17 > 23$ , $5 + 23 > 17$ , and $17 + 23 > 5$ .
Check First Set Calculations: Perform the calculations for the first set: $5 + 17 = 22$ , which is not greater than $23$ . Therefore, the first set \{$$5$,$17$,$23$\} cannot represent the sides of a triangle.
Apply Theorem to Second Set: Apply the Triangle Inequality Theorem to the second set ${6,20,25}$. Check if $6 + 20 > 25$, $6 + 25 > 20$, and $20 + 25 > 6$.
Check Second Set Calculations: Perform the calculations for the second set: $6 + 20 = 26$, which is greater than $25$. Now check the other two conditions.
Apply Theorem to Third Set: For the second set, check if $6 + 25 > 20$, which is $31 > 20$, and $20 + 25 > 6$, which is $45 > 6$. All conditions are satisfied, so the second set \{$6,20,25$\} could represent the sides of a triangle.
Check Third Set Calculations: Apply the Triangle Inequality Theorem to the third set ${5,17,24}$. Check if $5 + 17 > 24$, $5 + 24 > 17$, and $17 + 24 > 5$.
Apply Theorem to Fourth Set: Perform the calculations for the third set: $5 + 17 = 22$, which is not greater than $24$. Therefore, the third set \{$5,17,24$\} cannot represent the sides of a triangle.
Check Fourth Set Calculations: Apply the Triangle Inequality Theorem to the fourth set $\{10,13,25\}$. Check if $10 + 13 > 25$, $10 + 25 > 13$, and $13 + 25 > 10$.
Check Fourth Set Calculations: Apply the Triangle Inequality Theorem to the fourth set $\{10,13,25\}$. Check if $10 + 13 > 25$, $10 + 25 > 13$, and $13 + 25 > 10$. Perform the calculations for the fourth set: $10 + 13 = 23$, which is not greater than $25$. Therefore, the fourth set $\{10,13,25\}$ cannot represent the sides of a triangle.