In circle O, if chords AB and CD intersect at point E, what is the length of ED given AB = 11, AE = 9, and CE = 6?

To find the length of segment ED, we can use the properties of intersecting chords in a circle. When two chords intersect, the products of the lengths of the segments of each chord are equal.

In this case, we have chord AB, which is divided into segments AE and EB, and chord CD, which is divided into segments CE and ED. We know the following:

• The total length of chord AB is 11, and since AE is 9, we can find EB by subtracting AE from AB:

EB = AB - AE = 11 - 9 = 2.

Now we have:

• AE = 9

• EB = 2

• CE = 6

• Let ED = x (we need to find this length).

According to the property of intersecting chords, we have the equation:

AE × EB = CE × ED.

Substituting in the lengths we know:

9 × 2 = 6 × x.

This simplifies to:

18 = 6 × x.

To find x, we divide both sides by 6:

x = 18 / 6,

x = 3.

Thus, the length of segment ED is 3, making it the correct answer