Answer:
The length of the third side could be anywhere between [tex]7\; \rm in[/tex] and [tex]17\; \rm in[/tex], excluding the two endpoints.
Step-by-step explanation:
Let [tex]x[/tex] represent the length (in inches) of the third side.
In a triangle, the sum of the lengths of any two sides should be greater than the length of the third side. The lengths of the three sides of this triangle are:
- [tex]5\; \rm in[/tex],
- [tex]12\; \rm in[/tex], and
- [tex]x\; \rm in[/tex].
The sum of the lengths of the first two sides is [tex]5 + 12 = 17\; \rm in[/tex]. This sum should be greater than (not equal to) the length of the third side: [tex]x\; \rm in[/tex]. That gives the first inequality:
Similarly, the sum of the lengths of the first and the third sides is [tex](5 + x)\; \rm in[/tex]. This sum should be greater than (not equal to) the length of the second side: [tex]12\; \rm in[/tex]. That gives the second inequality:
The sum of the lengths of the second and the third sides is [tex](12 + x)\; \rm in[/tex]. This sum should be greater than (not equal to) the length of the first side: [tex]5\; \rm in[/tex]. That gives the third inequality:
Solve the three inequalities for the range of [tex]x[/tex], the length of the third side:
[tex]\displaystyle \left\lbrace\begin{aligned}& 5 + 12 > x \\ & 5 + x > 12 \\ & 12 + x > 5\end{aligned}\right.[/tex].
The first inequality simplifies to [tex]x < 17[/tex]. The second inequalities gives [tex]x > 7[/tex], while the third gives [tex]x > -7[/tex].
Refer to the diagram attached. [tex]7 < x < 17[/tex] is the region that satisfies all three inequalities. Therefore, the length of the third side should be between [tex]7[/tex] and [tex]17[/tex], exclusive.
