Question

Find the perimeter of the figure outlined by the solid line in terms of x.

Answer 1

Hint: Perimeter in simple words is just the length of the boundary. We can divide the diagram in small well-defined figures which have a direct formula for the perimeter. After finding the perimeter for each small figure we can add them up to get the perimeter as a whole.

Complete step-by-step answer:
Finding perimeter.
The given figure can be divided in $2$ part .
One of the parts is a semi-circle and the other one is a triangle.
We know that the perimeter of a circle is $2\pi r$
Perimeter of the triangle is the sum of all the sides.
Perimeter of circle:
$\begin{align}
  & =\pi r \\
 & =\pi 6x \\
 & =6\pi x \\
\end{align}$
Now, we need to find the perimeter of triangle.
For that we need to find the third side of triangle using the Pythagoras theorem.
$\begin{align}
  & c=\sqrt{{{a}^{2}}+{{b}^{2}}} \\
 & c=\sqrt{{{\left( 4x \right)}^{2}}+{{\left( 3x \right)}^{2}}} \\
 & \\
\end{align}$
 $\begin{align}
  & c=\sqrt{16x+9x} \\
 & c=5x \\
\end{align}$
Now ,
Perimeter
$\begin{align}
  & =5x+5x \\
 & =10x \\
\end{align}$

Total Perimeter$=10x+6\pi x$

Note: The other method to solve this question is to take more divisions in the original figure but that will increase unnecessary computation which we should avoid. Now, one more thing to be remembered here is that the question is asking only about the perimeter of the solid line and hence no internal sides to be included. We cannot include the base of the triangle and diameter of the semi-circle .Including them might give a wrong answer.