In the accompanying diagram of circle O, chord AB busects CD at E. If AE=8 and BE=9, find the length of CE in simplest radical form.
To find the length of CE in simplest radical form, we can use the property that states that if a chord bisects another chord, the products of the segments are equal.
In this problem, we are given that AE = 8 and BE = 9. Applying the property mentioned earlier, we can set up the following equation:
AE * BE = CE * DE
Substituting in the given values, we have:
8 * 9 = CE * DE
72 = CE * DE
Now, we need to find the value of DE. To do that, we can use the property that states that the product of the segments of a chord and its external part is equal to the square of the radius of the circle. In other words:
DE * (DE + CE) = r^2
Where r is the radius of the circle, and in this case, it is half the length of CD (since CD is a diameter). Thus, r = CD/2.
Substituting CD/2 for r, we have:
DE * (DE + CE) = (CD/2)^2
By substituting 72/CE for DE (from the previous equation), we can solve for CE:
(72/CE) * ((72/CE) + CE) = (CD/2)^2
Simplifying the equation:
72 * (72 + CE^2) = CD^2 / 4
5184 + 72CE^2 = CD^2 / 4
Multiplying both sides by 4:
20736 + 288CE^2 = CD^2
Since we're asked to find the length of CE in simplest radical form, we'll leave the equation in this form for now.
Unfortunately, there is not enough information provided in the problem to determine the exact value of CE.