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.

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....

$(AC)^2+(AB)^2= (BC)^2\\ \\ h^2+(12)^2= (13)^2\\ \\ h^2+144= 169\\ \\ h^2= 25\\ \\ h= \sqrt{25}= 5$

So, the length of that altitude is 5 cm.