Question

1. in the accompanying diagram of circle o, chord \\( \overline { c d } \\) bisects chord \\( \overline { a b } \\) at e, \\( c e = 2 \\), and \\( a b = 8 \\). find ed.

Explanation:

Step1: Apply the Chord - Chord Product Theorem

If two chords \(AB\) and \(CD\) intersect at a point \(E\) inside a circle, then \(AE\times EB=CE\times ED\).
Since chord \(CD\) bisects chord \(AB\) at \(E\) and \(AB = 8\), then \(AE=EB=\frac{AB}{2}\). So \(AE = EB=4\), and \(CE = 2\). Let \(ED=x\).

Step2: Substitute values into the formula

Substitute \(AE = 4\), \(EB = 4\), and \(CE = 2\) into the equation \(AE\times EB=CE\times ED\). We get \(4\times4=2\times x\).

Step3: Solve for \(x\)

The equation \(16 = 2x\) can be solved by dividing both sides by 2. So \(x=\frac{16}{2}=8\).

Answer:

\(ED = 8\)