Question

8. Find the perimeter of CDEF. Assume that all segments that appear to be tangent.

Answer 1

In order to solve tthis problem we must apply the "tangent quadrilateral theorem" for inscribed circles.

The theorem states the following:

$\bar{AB}+\bar{CD}=\bar{BC}+\bar{DA}$

In our case we have:

From the question we see that 7.4 is the radius of the circle. So the sides of the quadrilateral are:

CF = 13

FE = 12.1

ED = 7.4 + 14 = 21.4

CD = 7.4 + x

As we see, we don't know the value of x. But we can apply the "tangent quadrilateral theorem" for inscribed circles.

Which says in this case that:

$\begin{gathered} CF+ED=CD+FE \\ 13+21.4=(7.4+x)+12.1 \\ 34.4=x+19.5 \\ x=34.4-19.5=14.9 \end{gathered}$

Now we can calculate the perimeter of CDEF summing the sides:

$\text{CDEF}=CD+ED+FE+CF=(7.4+14.9)+21.4+12.1+13=68.8$