Question

given the circle below with secants \\( \overline { a b c } \\) and \\( \overline { e d c } \\), find the length of \\( \overline { a b } \\). round to the nearest tenth if n

Explanation:

Step1: Apply the secant - secant theorem

The secant - secant theorem states that if two secants \(ABC\) and \(EDC\) are drawn to a circle from an external point \(C\), then \(CA\times CB=CE\times CD\). Let \(AB = x\), then \(CA=x + 15\), \(CB = 15\), \(CE=22+18=40\), and \(CD = 18\).
So, \((x + 15)\times15=40\times18\).

Step2: Expand and solve the equation

Expand the left - hand side: \(15x+225 = 720\).
Subtract 225 from both sides: \(15x=720 - 225\).
\(15x=495\).
Divide both sides by 15: \(x=\frac{495}{15}\).

Answer:

\(33\)