Question

given the circle below with secants \\( \overline{k l m} \\) and \\( \overline{o n m} \\). if \\( n m=11, o n=27 \\) and \\( l m=12 \\), find the length of \\( \overline{k l} \\). round to the nearest tenth if necessary.

Explanation:

Step1: Apply the secant - secant rule

The secant - secant rule states that if two secants \(M K L\) and \(M O N\) are drawn to a circle from an external point \(M\), then \(ML\times MK=MN\times MO\). Let \(KL = x\). Then \(MK=ML + KL=12 + x\), \(MN = 11\), and \(MO=MN + ON=11 + 27=38\).
So, the equation is \(11\times38=12\times(12 + x)\).

Step2: Simplify the equation

First, calculate \(11\times38 = 418\) and \(12\times(12 + x)=144+12x\).
The equation becomes \(418=144 + 12x\).
Subtract 144 from both sides: \(418-144=12x\), so \(274 = 12x\).

Step3: Solve for \(x\)

Divide both sides by 12: \(x=\frac{274}{12}\approx22.8\).

Answer:

\(22.8\)