Answer: The correct option is
(a) 6 cm.
Step-by-step explanation: Given that two sides of a triangle have lengths 15 cm and 19 cm.
We are to select the correct measurement that can be the length of the third side.
Let x units represents the length of the third side.
We know that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
So, we have
![x+15>19\\\\\Rightarrow x>19-15\\\\\Rightarrow x>4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)](https://tex.z-dn.net/?f=x%2B15%3E19%5C%5C%5C%5C%5CRightarrow%20x%3E19-15%5C%5C%5C%5C%5CRightarrow%20x%3E4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%28i%29)
![x+19>15\\\\\Rightarrow x>15-19\\\\\Rightarrow x>-4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)](https://tex.z-dn.net/?f=x%2B19%3E15%5C%5C%5C%5C%5CRightarrow%20x%3E15-19%5C%5C%5C%5C%5CRightarrow%20x%3E-4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%28ii%29)
and
![15+19>x\\\\\Rightarrow x](https://tex.z-dn.net/?f=15%2B19%3Ex%5C%5C%5C%5C%5CRightarrow%20x%3C34~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%28iii%29)
Combining inequalities (i), (ii) and (iii), we get
![4](https://tex.z-dn.net/?f=4%3Cx%3C34.)
Therefore, from the given options, only 6 lies between 4 and 34.
Thus, the correct option is (a) 6 cm.