Question

given the circle below with secant \\( \overline { m l k } \\) and tangent \\( \overline { j k } \\), find the length of \\( \overline { m l } \\). round to the nearest tenth if necessary.

Explanation:

Step1: Use the tangent - secant theorem

The tangent - secant theorem states that if \(JK\) is a tangent and \(MK\) is a secant of a circle, then \(JK^{2}=KL\times KM\). Let \(ML = x\), then \(KM=KL + ML=10 + x\). Given \(JK = 14\) and \(KL = 10\).
Substitute into the formula: \(14^{2}=10\times(10 + x)\).

Step2: Solve the equation

First, expand the right - hand side: \(196=100 + 10x\).
Then, subtract 100 from both sides: \(196−100 = 10x\), so \(96 = 10x\).
Finally, solve for \(x\): \(x=\frac{96}{10}=9.6\).

Answer:

\(9.6\)