What are possible values for the third side of a triangle with side lengths of 8 cm and 10 cm to make it a right triangle?

To determine the possible lengths for the third side of a triangle that would make it a right triangle, we can use the Pythagorean theorem. For a triangle with sides ( a ), ( b ), and ( c ), where ( c ) is the longest side, the relationship is:

[ a^2 + b^2 = c^2 ]

Given two sides of lengths 8 cm and 10 cm, we will check which of the provided options can be the third side.

1. 6 cm:

– ( 8^2 + 6^2 = 64 + 36 = 100 )

– ( 10^2 = 100 )

– This is valid.

2. 18 cm:

– ( 10^2 + 8^2 = 64 + 100 )

– This would not hold since ( 10^2 + 8^2 = 164 neq 18^2 ).

– This is not valid.

3. 2 cm:

– ( 8^2 + 2^2 = 64 + 4 = 68 )

– ( 10^2 = 100 )

– This is not valid.

4. √164 cm:

– ( 8^2 + 10^2 = 64 + 100 = 164 )

– This