Question

One of the angles of a parallelogram is ${150}^{\circ }$. Altitudes are drawn from the vertex of this angle. If these altitudes measure $6$ cm and $8$ cm, then find the perimeter of the parallelogram.

Answer 1

Given- $ABCD$is a parallelogram with $\angle BAD={150}^o$

Altitudes $CF$ on $AB$ and $AE$ on $DC$ are $6$ cm and $8$ cm respectively.

To find out- the perimeter of $ABCD$=?

Solution:

Given that $\angle BAD={150}^o$

Since adjacent angles of a parallelogram are supplimentary, we get

$\angle ABC={180}^o-\angle BAD={180}^o-{150}^o={30}^o$

Also $CF=6$ cm

$\therefore \frac{CF}{BC}=\sin {30}^o$

$\Rightarrow BC=\frac{BC}{\sin {30}^o}=\frac{6}{\frac{1}{2}}$ cm $=12$ cm and $AE=8$ cm given.....(i)

Now $arABCD=$ base $\times$ height $=AB\times CF=BC\times AE$

$\Rightarrow AB=\frac{BC\times AE}{CF}=\frac{12\times 8}{6}$ cm $=16$ cm ....(from i)

$\therefore$ the perimeter $\left(ABCD\right)=2(AB+BC)=2(16+12)cm=56$ cm