Question

given the inequality 2 < x < 18 to describe all possible side lengths for the third side of the triangle, which are the other two side lengths?
10 and 8
2 and 18
7 and 11
2 and 16

Explanation:

Step1: Recall triangle - inequality theorem

The length of the third side $x$ of a triangle with side lengths $a$ and $b$ satisfies $|a - b|\lt x\lt a + b$.

Step2: Check option A

For $a = 10$ and $b = 8$, $|10 - 8|=2$ and $10 + 8 = 18$, so $2\lt x\lt18$.

Step3: Check option B

For $a = 2$ and $b = 18$, $|2 - 18| = 16$ and $2+18 = 20$, the inequality for the third - side is $16\lt x\lt20$.

Step4: Check option C

For $a = 7$ and $b = 11$, $|7 - 11| = 4$ and $7 + 11 = 18$, the inequality for the third - side is $4\lt x\lt18$.

Step5: Check option D

For $a = 2$ and $b = 16$, $|2 - 16| = 14$ and $2+16 = 18$, the inequality for the third - side is $14\lt x\lt18$.

Answer:

A. 10 and 8