Which of the following sets of numbers could represent the three sides of a triangle? $\newline$ $\{12,16,29\}$ $\newline$ $\{4,11,17\}$ $\newline$ $\{12,25,38\}$ $\newline$ $\{7,10,16\}$

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Domain and range of absolute value functions: equations

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

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\{12,16,29\}

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\{4,11,17\}

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\{12,25,38\}

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\{7,10,16\}

Triangle Inequality Theorem Application: To determine if a set of numbers can represent the sides of a triangle, we use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let's apply this to the first set $\{12, 16, 29\}$ . We check if 12 + 16 > 29, 12 + 29 > 16, and 16 + 29 > 12. $12 + 16 = 28$ , which is not greater than $29$ .
First Set Analysis: Since $12 + 16$ is not greater than $29$ , the first set of numbers \{ $12, 16, 29$ \} cannot represent the sides of a triangle.
Second Set Analysis: Now let's check the second set ${4, 11, 17}$ . We check if 4 + 11 > 17, 4 + 17 > 11, and 11 + 17 > 4. $4 + 11 = 15$ , which is not greater than $17$ .
Third Set Analysis: Since $4 + 11$ is not greater than $17$ , the second set of numbers \{ $4, 11, 17$ \} cannot represent the sides of a triangle.
Fourth Set Analysis: Next, we check the third set ${12, 25, 38}$ . We check if 12 + 25 > 38, 12 + 38 > 25, and 25 + 38 > 12. $12 + 25 = 37$ , which is not greater than $38$ .
Fourth Set Analysis: Next, we check the third set ${12, 25, 38}$ . We check if 12 + 25 > 38, 12 + 38 > 25, and 25 + 38 > 12. $12 + 25 = 37$ , which is not greater than $38$ .Since $12 + 25$ is not greater than $38$ , the third set of numbers ${12, 25, 38}$ cannot represent the sides of a triangle.
Fourth Set Analysis: Next, we check the third set ${12, 25, 38}$ . We check if 12 + 25 > 38, 12 + 38 > 25, and 25 + 38 > 12. $12 + 25 = 37$ , which is not greater than $38$ .Since $12 + 25$ is not greater than $38$ , the third set of numbers ${12, 25, 38}$ cannot represent the sides of a triangle.Finally, let's check the fourth set ${7, 10, 16}$ . We check if 12 + 25 > 38 $0$ , 12 + 25 > 38 $1$ , and 12 + 25 > 38 $2$ . 12 + 25 > 38 $3$ , which is greater than 12 + 25 > 38 $4$ . 12 + 25 > 38 $5$ , which is greater than 12 + 25 > 38 $6$ . 12 + 25 > 38 $7$ , which is greater than 12 + 25 > 38 $8$ . All conditions are satisfied.