The length of that altitude is 5 cm.
Explanation
According to the below diagram, [tex]ABCD[/tex] is a parallelogram with diagonal [tex]\overline{AC}[/tex] as its altitude.
Suppose, the length of side [tex]\overline{AB}[/tex] is [tex]x[/tex] cm.
As the length of one side is 1 cm longer than the length of the other, so the length of side [tex]\overline{BC}[/tex] will be: [tex](x+1) cm[/tex]
Given that, the perimeter of the parallelogram is 50 cm. So, the equation will be.....
[tex]2[x+(x+1)]=50\\ \\ 2(2x+1)=50\\ \\ 4x+2=50\\ \\ 4x=48\\ \\ x= 12[/tex]
So, the length of [tex]\overline{AB}[/tex] is 12 cm and the length of [tex]\overline{BC}[/tex] is (12+1)= 13 cm.
Suppose, the length of the altitude([tex]\overline{AC}[/tex]) is [tex]h[/tex] cm.
Now, in right angle triangle [tex]ABC[/tex], using Pythagorean theorem....
[tex](AC)^2+(AB)^2= (BC)^2\\ \\ h^2+(12)^2= (13)^2\\ \\ h^2+144= 169\\ \\ h^2= 25\\ \\ h= \sqrt{25}= 5[/tex]
So, the length of that altitude is 5 cm.
