For each set of three lengths, determine if they can be the side lengths of a triangle. $\newline$ \begin{tabular}{|c|c|c|} $\newline$ \hline Lengths & \begin{tabular}{c} $\newline$ Can be side lengths \\ $\newline$ of a triangle $\newline$ \end{tabular} & \begin{tabular}{c} $\newline$ Cannot be side \\ $\newline$ lengths of a triangle $\newline$ \end{tabular} \\ $\newline$ \hline $7,9,7$ & & \\ $\newline$ \hline $22,8,17$ & & \\ $\newline$ \hline $6,14,11$ & & \\ $\newline$ \hline $9.7,15.8,6.6$ & & \\ $\newline$ \hline $\newline$ \end{tabular}

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Calculate mean, median, mode, and range

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Q. For each set of three lengths, determine if they can be the side lengths of a triangle.

\newline

\begin{tabular}{|c|c|c|}

\newline

\hline Lengths & \begin{tabular}{c}

\newline

Can be side lengths \\

\newline

of a triangle

\newline

\end{tabular} & \begin{tabular}{c}

\newline

Cannot be side \\

\newline

lengths of a triangle

\newline

\end{tabular} \\

\newline

\hline

7,9,7

& & \\

\newline

\hline

22,8,17

& & \\

\newline

\hline

6,14,11

& & \\

\newline

\hline

9.7,15.8,6.6

& & \\

\newline

\hline

\newline

\end{tabular}

Triangle Inequality Theorem: To check if three lengths can form a triangle, use the Triangle Inequality Theorem. The sum of any two sides must be greater than the third side.
Check Set $1$ : $7, 9, 7$ : Check the first set: $7, 9, 7$ . Add the two smallest lengths: $7 + 7 = 14$ . Is $14$ greater than the third side, $9$ ? Yes, $14 > 9$ .
Check Set $2$ : $22, 8, 17$ : Now, add the other two combinations: $7 + 9 = 16$ , and $9 + 7 = 16$ . Are these sums greater than the remaining side? Yes, $16 > 7$ for both.
Check Set $3$ : $6, 14, 11$ : Since all combinations satisfy the Triangle Inequality Theorem, the lengths $7, 9, 7$ can form a triangle.
Check Set $4$ : $9.7$ , $15.8$ , $6.6$ : Check the second set: $22$ , $8$ , $17$ . Add the two smallest lengths: $8 + 17 = 25$ . Is $25$ greater than the third side, $22$ ? Yes, $25 > 22$ .
Check Set $4$ : $9.7$ , $15.8$ , $6.6$ : Check the second set: $22$ , $8$ , $17$ . Add the two smallest lengths: $8 + 17 = 25$ . Is $25$ greater than the third side, $22$ ? Yes, $25 > 22$ .Now, add the other two combinations: $15.8$ $0$ , and $15.8$ $1$ . Are these sums greater than the remaining side? Yes, $15.8$ $2$ and $15.8$ $3$ .
Check Set $4$ : $9.7$ , $15.8$ , $6.6$ : Check the second set: $22$ , $8$ , $17$ . Add the two smallest lengths: $8 + 17 = 25$ . Is $25$ greater than the third side, $22$ ? Yes, $25 > 22$ .Now, add the other two combinations: $15.8$ $0$ , and $15.8$ $1$ . Are these sums greater than the remaining side? Yes, $15.8$ $2$ and $15.8$ $3$ .Since all combinations satisfy the Triangle Inequality Theorem, the lengths $22$ , $8$ , $17$ can form a triangle.
Check Set $4$ : $9.7, 15.8, 6.6$ : Check the second set: $22, 8, 17$ . Add the two smallest lengths: $8 + 17 = 25$ . Is $25$ greater than the third side, $22$ ? Yes, $25 > 22$ .Now, add the other two combinations: $22 + 8 = 30$ , and $22 + 17 = 39$ . Are these sums greater than the remaining side? Yes, $30 > 17$ and $39 > 8$ .Since all combinations satisfy the Triangle Inequality Theorem, the lengths $22, 8, 17$ can form a triangle.Check the third set: $22, 8, 17$ $1$ . Add the two smallest lengths: $22, 8, 17$ $2$ . Is $22, 8, 17$ $3$ greater than the third side, $22, 8, 17$ $4$ ? Yes, $22, 8, 17$ $5$ .
Check Set $4$ : $9.7, 15.8, 6.6$ : Check the second set: $22, 8, 17$ . Add the two smallest lengths: $8 + 17 = 25$ . Is $25$ greater than the third side, $22$ ? Yes, $25 > 22$ .Now, add the other two combinations: $22 + 8 = 30$ , and $22 + 17 = 39$ . Are these sums greater than the remaining side? Yes, $30 > 17$ and $39 > 8$ .Since all combinations satisfy the Triangle Inequality Theorem, the lengths $22, 8, 17$ can form a triangle.Check the third set: $22, 8, 17$ $1$ . Add the two smallest lengths: $22, 8, 17$ $2$ . Is $22, 8, 17$ $3$ greater than the third side, $22, 8, 17$ $4$ ? Yes, $22, 8, 17$ $5$ .Now, add the other two combinations: $22, 8, 17$ $6$ , and $22, 8, 17$ $7$ . Are these sums greater than the remaining side? Yes, $22, 8, 17$ $8$ and $22, 8, 17$ $9$ .
Check Set $4$ : $9.7$ , $15.8$ , $6.6$ : Check the second set: $22$ , $8$ , $17$ . Add the two smallest lengths: $8 + 17 = 25$ . Is $25$ greater than the third side, $22$ ? Yes, $25 > 22$ .Now, add the other two combinations: $15.8$ $0$ , and $15.8$ $1$ . Are these sums greater than the remaining side? Yes, $15.8$ $2$ and $15.8$ $3$ .Since all combinations satisfy the Triangle Inequality Theorem, the lengths $22$ , $8$ , $17$ can form a triangle.Check the third set: $15.8$ $7$ , $15.8$ $8$ , $15.8$ $9$ . Add the two smallest lengths: $6.6$ $0$ . Is $17$ greater than the third side, $15.8$ $8$ ? Yes, $6.6$ $3$ .Now, add the other two combinations: $6.6$ $4$ , and $6.6$ $5$ . Are these sums greater than the remaining side? Yes, $6.6$ $6$ and $6.6$ $7$ .Since all combinations satisfy the Triangle Inequality Theorem, the lengths $15.8$ $7$ , $15.8$ $8$ , $15.8$ $9$ can form a triangle.
Check Set $4$ : $9.7$ , $15.8$ , $6.6$ : Check the second set: $22$ , $8$ , $17$ . Add the two smallest lengths: $8 + 17 = 25$ . Is $25$ greater than the third side, $22$ ? Yes, $25 > 22$ .Now, add the other two combinations: $15.8$ $0$ , and $15.8$ $1$ . Are these sums greater than the remaining side? Yes, $15.8$ $2$ and $15.8$ $3$ .Since all combinations satisfy the Triangle Inequality Theorem, the lengths $22$ , $8$ , $17$ can form a triangle.Check the third set: $15.8$ $7$ , $15.8$ $8$ , $15.8$ $9$ . Add the two smallest lengths: $6.6$ $0$ . Is $17$ greater than the third side, $15.8$ $8$ ? Yes, $6.6$ $3$ .Now, add the other two combinations: $6.6$ $4$ , and $6.6$ $5$ . Are these sums greater than the remaining side? Yes, $6.6$ $6$ and $6.6$ $7$ .Since all combinations satisfy the Triangle Inequality Theorem, the lengths $15.8$ $7$ , $15.8$ $8$ , $15.8$ $9$ can form a triangle.Check the fourth set: $9.7$ , $15.8$ , $6.6$ . Add the two smallest lengths: $22$ $4$ . Is $22$ $5$ greater than the third side, $15.8$ ? No, $22$ $5$ is not greater than $15.8$ .