Question

In a parallelogram $ABCD$, $AB=4x$ and $AD=2x+1$. If the perimeter is $38cm$ and area is $60{cm}^2$, find the length of the altitude from $D$ on to $AB$.

Answer 1

Perimeter of a parallelogram is $2(l+b)$ where $l$ is the length and $b$ is the breadth of the parallelogram.

It is given that the perimeter of the parallelogram is $P=38$ cm, therefore,

$2(l+b)=38$ implies

$l+b=19$

Now, it is also given that $AB=4x$ and $AD=2x+1$, thus substitute $l=4x$ and $b=2x+1$ in $l+b=19$ as follows:

$4x+2x+1=19$

$6x=19–1$

$6x=18$

$x=3$

Therefore, $AB=4x=4\times 3=12$ cm.

Let the altitude $DE=h$, then the given area is $60$ cm${}^2$ therefore,

$12\times h=60$

$\Rightarrow h=\frac{60}{12}=5$

Hence, the length of the altitude from $D$ on to $AB$ is $5$ cm.