# Triangle has two sides that have lengths of 17 feet and 22 feet. Which ofthe following lengths could not represent the length of the third side?

Dec 10, 2016

length of the third side will have a value between 5 and 39.

#### Explanation: Sol. 1)
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let $c$ be the third side.
Given that $a = 17 , \mathmr{and} b = 22$,
i) a+c>b, => 17+c>22, => c>5
ii) a+b>c, => 17+22>c, => 39>c
$\implies 5 < c < 39$
Hence, the third side $c$ will have a value between $5$ and $39$.

Sol.2)
Using the cosine law of solving triangles.
Let the included angle between sides $a \mathmr{and} b$ be $x$.
${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos x$
${c}^{2} = {17}^{2} + {22}^{2} - 2 \times 17 \times 22 \times \cos x$
${c}^{2} = 773 - 748 \cos x$
as ${0}^{\circ} < x < {180}^{\circ} , \mathmr{and} - 1 < \cos x < 1$,
$\implies \sqrt{773 - 748} < c < \sqrt{773 + 748}$
or $\sqrt{25} < c < \sqrt{1521}$
or $5 < c < 39$
Hence, the third side $c$ will have a value between $5$ and $39$.