Question

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. The lengths of two sides of a triangle are $16\mathrm{ft}$ and $21\mathrm{ft}$. Find the possible lengths of the third side?

Answer 1

Find the possible lengths of the third side:

Let the length of the third side be $\mathrm{x}\mathrm{ft}$.

Then according to the Triangle Inequality,

$\bullet$ $\mathrm{x}+16>21,\mathrm{x}>0$

$\Rightarrow$ $\mathrm{x}>5$ $...\left(\mathrm{i}\right)$

$\bullet$$\mathrm{x}+21>16,\mathrm{x}>0$

$\Rightarrow$ True for any $\mathrm{x}>0$ $...\left(\mathrm{ii}\right)$

$\bullet$$16+21>\mathrm{x},\mathrm{x}>0$

$\Rightarrow$ $37>\mathrm{x},\mathrm{x}>0$ $...\left(\mathrm{iii}\right)$

From $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$, we get; $5\mathrm{ft}<\mathrm{x}<37\mathrm{ft}$

Hence, the possible lengths of the third side is $>5\mathrm{ft}$ and $<37\mathrm{ft}$.