Question

in the accompanying diagram of circle o, chords \\(\overline{ab}\\) and \\(\overline{cd}\\) intersect at e. if \\(ae = 4\\), \\(eb = 6\\), and \\(ce = 3\\), find \\(ed\\).

Explanation:

Step1: Recall the Intersecting Chords Theorem

The Intersecting Chords Theorem states that if two chords intersect at a point inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. Mathematically, if chords \( \overline{AB} \) and \( \overline{CD} \) intersect at \( E \), then \( AE \times EB = CE \times ED \).

Step2: Substitute the given values into the formula

We are given \( AE = 4 \), \( EB = 6 \), and \( CE = 3 \). Substituting these values into the formula \( AE \times EB = CE \times ED \), we get:
\[
4 \times 6 = 3 \times ED
\]

Step3: Solve for \( ED \)

First, calculate the left - hand side of the equation: \( 4\times6 = 24 \). So the equation becomes:
\[
24 = 3\times ED
\]
To solve for \( ED \), divide both sides of the equation by 3:
\[
ED=\frac{24}{3}=8
\]

Answer:

\( 8 \)