Question

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?

Answer 1

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, and b=22`,
`i) a+c>b, => 17+c>22, => c>5`
`ii) a+b>c, => 17+22>c, => 39>c`
`=> 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 and b` be `x`.
`c^2=a^2+b^2-2abcosx`
`c^2=17^2+22^2-2xx17xx22xxcosx`
`c^2=773-748cosx`
as `0^@ < x < 180^@, and -1 < cosx < 1`,
`=> sqrt(773-748) < c < sqrt(773+748)`
or `sqrt25 < c < sqrt1521`
or `5 < c < 39`
Hence, the third side `c` will have a value between `5` and `39`.