Full solution

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.