Take a look at this polygon circumscribing the circle. Let's begin by naming the vertices and points of tangency on the given diagram.
Notice that in order to find the perimeter, we need to find the length of EG, which consists of two segments, EF and FG. First, recall that if two segments to a share a common endpoint outside the circle, then the segments are .This means that CB and CD are the same length.
The length of CE is 12. Therefore, we can find the length of DE by subtracting 6, which is the length of CD, from 12.
DE&= 12-6
&= 6
Now that we know the length of DE, we can use the same fact as before to find the length of EF.
Again, we also know that the lengths of AB and AJ are the same. Since the length of AB is equal to 7, so is the length of AJ.
Let's now focus on the length of IJ. By subtracting the length of AJ from the length of AI we find that the length of IJ is 10.
We will once again use the fact that two segments tangent to a circle that share a common endpoint are congruent. This gives us that the length of HI is also 10.
Since the length of GI=15, we can find the length of GH by subtracting the length of HI from 15.
GH&= 15- 10
&=5
Again, due to the same fact as before, we know that the length of FG is 5 as well.
Now that we know all the side lengths, we can add them in order to find the perimeter of the circumscribed polygon.
&P= 6+ 6+6+6+ 7+ 17+ 10+ 10+5+5
&P=68 cm
