Question

1. in the accompanying diagram of circle o, chords ab and cd intersect at point e. if ae = 2, cd = 9, and ce = 4, find be.
2. two secants, abc and ade, are drawn to a circle from external point a. if ab = 4, bc = 6, and ad = 5, find de.

Explanation:

Step1: Find the length of \(DE\)

According to the intersecting - chords theorem, if two chords \(AB\) and \(CD\) intersect at a point \(E\) inside a circle, then \(AE\times BE=CE\times DE\).
We know that \(CD = 9\) and \(CE = 4\), so \(DE=CD - CE=9 - 4 = 5\).

Step2: Substitute values into the formula

We are given \(AE = 2\), \(CE = 4\), \(DE = 5\), and we need to find \(BE\).
Using the formula \(AE\times BE=CE\times DE\), we substitute the values: \(2\times BE=4\times5\).

Step3: Solve for \(BE\)

From \(2\times BE = 20\), we can solve for \(BE\) by dividing both sides of the equation by \(2\). So \(BE=\frac{4\times5}{2}\).

Answer:

\(BE = 10\)