Question

The lengths of the sides of a triangle are 12, 13, and n. Which of the following must be true? n≥1 n<13 1

Answer 1

`\begin{cases} 12+13 \ \textgreater \ n \\ 12+n \ \textgreater \ 13 \\ 13+n \ \textgreater \ 12 \end{cases} \\ \\ \begin{cases} n \ \textless \ 25 \\ n \ \textgreater \ 1 \\ n \ \textgreater \ -1 \end{cases} \\ \\ n\in (1,25)`

Answer 2

D. 1 < n < 25

Explanation:

Choices for this question are

( A ) n greater than or equal to 1

( B ) n < 13

( C ) 1 < n < 13

( D ) 1 < n < 25

The triangular inequality theorem states that the sum of two sides of a triangle must be greater than the third side. That means

`12+13>n\\25>n\\n<25`

Also, that third side most be greater than 1, other wise, the side wouldn't exist.

Therefore, according to the triangular inequality theorem, the best answer is D. 1 < n < 25