Can a triangle have sides with the lengths of 4cm, 5cm and 9cm? Why or why not?

Question

Can a triangle have sides with the lengths of 4cm, 5cm and 9cm? Why or why not?

Answer 1

No because the sum of the two shorter sides must exceed the third side.

Answer 2

Hi Breanna,

 

Imagine that the 4 cm leg and the 5 cm leg were joined together at an angle θ. We would like to place a third leg c across from the angle θ. The Law of Cosines can be used to give a relationship among all of the stated quantities:

 

c² = 4² + 5² - 2(4)(5)cosθ

 

We know that c = 9, from the given. So,

 

9² = 4² + 5² - 2(4)(5)cosθ

 

or

 

81 = 16 + 25 - 40cosθ

 

or

 

40cosθ = -40

 

cosθ = -1

 

So,

 

θ = 180°

 

Physically, this is a pathological case. This means that the 4 cm and 5 cm legs would lie flat against the 9 cm leg. So, this is not a triangle.