Question

Two sides of a trangie measure 5 in and 12 in which could be the length of the third side?

Answer 1

The length of the third side could be anywhere between
`7\; \rm in` and
`17\; \rm in`, excluding the two endpoints.

Explanation:

Let
`x` 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:

• 
  `5\; \rm in`,
• 
  `12\; \rm in`, and
• 
  `x\; \rm in`.

The sum of the lengths of the first two sides is
`5 + 12 = 17\; \rm in`. This sum should be greater than (not equal to) the length of the third side:
`x\; \rm in`. That gives the first inequality:

• 
  `5 + 12 > x`.

Similarly, the sum of the lengths of the first and the third sides is
`(5 + x)\; \rm in`. This sum should be greater than (not equal to) the length of the second side:
`12\; \rm in`. That gives the second inequality:

• 
  `5 + x > 12`.

The sum of the lengths of the second and the third sides is
`(12 + x)\; \rm in`. This sum should be greater than (not equal to) the length of the first side:
`5\; \rm in`. That gives the third inequality:

• 
  `12 + x> 5`.

Solve the three inequalities for the range of
`x`, the length of the third side:

`\displaystyle \left\lbrace\begin{aligned}& 5 + 12 > x \\ & 5 + x > 12 \\ & 12 + x > 5\end{aligned}\right.`.

The first inequality simplifies to
`x < 17`. The second inequalities gives
`x > 7`, while the third gives
`x > -7`.

Refer to the diagram attached.
`7 < x < 17` is the region that satisfies all three inequalities. Therefore, the length of the third side should be between
`7` and
`17`, exclusive.