Question

given the circle below with secants (overline{klm}) and (overline{onm}), find the length of (overline{kl}). round to the nearest tenth if necessary.

Explanation:

Step1: Apply the secant - secant theorem

The secant - secant theorem states that if two secants \( \overline{KLM}\) and \( \overline{ONM}\) are drawn to a circle from an external point \(M\), then \(ML\times MK=MN\times MO\). Let \(KL = x\). Then \(MK=ML + KL=22 + x\), \(MN = 21\), and \(MO=21 + 27=48\).
Substitute into the formula: \(22(22 + x)=21\times48\).

Step2: Expand and solve the equation

Expand the left - hand side: \(22\times22+22x=21\times48\).
\(484+22x = 1008\).
Subtract \(484\) from both sides: \(22x=1008 - 484\).
\(22x=524\).
Divide both sides by \(22\): \(x=\frac{524}{22}\approx23.8\).

Answer:

\(23.8\)