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?

Question

Sadie Allen · Accepted Answer

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

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#.

Claire Akin · Answer

undefined According to the triangle inequality theorem, the length of the third side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference.

For a triangle with sides of 17 feet and 22 feet:
- The third side cannot be greater than 39 feet (17 + 22).
- The third side cannot be less than 5 feet (|17 - 22|).

So, any length less than 5 feet or greater than 39 feet could not represent the length of the third side.