Question

One of diagonals of a parallelogram is its altitude. What is the length of this altitude, if its perimeter is 50 cm, and the length of one side is 1 cm longer than the length of the other?

Answer 1

The length of that altitude is 5 cm.

Step-by-step explanation

According to the below diagram,
`ABCD` is a parallelogram with diagonal
`\overline{AC}` as its altitude.

Suppose, the length of side
`\overline{AB}` is
`x` cm.

As the length of one side is 1 cm longer than the length of the other, so the length of side
`\overline{BC}` will be:
`(x+1) cm`

Given that, the perimeter of the parallelogram is 50 cm. So, the equation will be.....

`2[x+(x+1)]=50\\ \\ 2(2x+1)=50\\ \\ 4x+2=50\\ \\ 4x=48\\ \\ x= 12`

So, the length of
`\overline{AB}` is 12 cm and the length of
`\overline{BC}` is (12+1)= 13 cm.

Suppose, the length of the altitude(
`\overline{AC}`) is
`h` cm.

Now, in right angle triangle
`ABC`, using Pythagorean theorem....

`(AC)^2+(AB)^2= (BC)^2\\ \\ h^2+(12)^2= (13)^2\\ \\ h^2+144= 169\\ \\ h^2= 25\\ \\ h= √(25)= 5`

So, the length of that altitude is 5 cm.