To solve for the length of DE and CE, we can use the triangle proportionality theorem and the properties of similar triangles.
a) To find the length of DE:
We can see that triangle ABC and triangle ADE are similar triangles since they have the same angles. Using the triangle proportionality theorem, we can set up the following proportion:
AB/AD = BC/DE
Substituting the given values, we have:
8/AD = 9/DE
To solve for DE, we can cross-multiply and then solve for DE:
8 * DE = 9 * AD
DE = (9 * AD) / 8
b) To find the length of CE:
We can see that triangle BCD and triangle CED are similar triangles since they have the same angles. Using the triangle proportionality theorem, we can set up the following proportion:
CD/CE = BD/DE
Substituting the given values, we have:
9/CE = 4/DE
We already found the value of DE in part a, so we can substitute that value into the equation:
9/CE = 4/((9 * AD) / 8)
To solve for CE, we can cross-multiply and then solve for CE:
9 * ((9 * AD) / 8) = 4 * CE
CE = (81 * AD) / 32
To find the lengths of DE and CE, you would need to know the value of AD. If you have that information, you can substitute it into the equations we derived in parts a and b to find the lengths of DE and CE, respectively.