a) CE = 12
b) DE = 8
c) FH= 8
d) Perimeter of Triangle CDE= 44
a)Finding CE:
We are given that `FG = 9` and `CD = 24`. Since `FG` is parallel to `CD` and is a midsegment of `AB`, we can apply the Triangle Midsegment Theorem. This theorem states that a midsegment of a triangle is parallel to the third side and half its length. Therefore, we can find the length of `CE` as follows:
`CE = (1/2) * CD = (1/2) * 24 = 12`
b)Finding DE:
Similarly, we know that `DE` is parallel to `AB` and is half its length. We can find the length of `AB` by adding the lengths of the two segments it is divided into by the midsegment `FG`:
`AB = FG + GH = 9 + 7 = 16`
Therefore, the length of `DE` is:
`DE = (1/2) * AB = (1/2) * 16 = 8`
c)Finding FH:
We can also find the length of `FH` using the Triangle Midsegment Theorem. As mentioned earlier, a midsegment is half the length of the side it is parallel to. In this case, `FH` is parallel to `AB`, so we can find its length as follows:
`FH = (1/2) * AB = (1/2) * 16 = 8`
d)Perimeter of Triangle CDE:
Finally, to find the perimeter of triangle `CDE`, we simply add the lengths of its three sides:
`Perimeter of triangle CDE = CE + DE + CD = 12 + 8 + 24 = 44`