Question

1. given the circle below with tangent \\( \overline{j k} \\) and secant \\( \overline{m k} \\). if \\( m l=10 \\) and \\( m k=12 \\), find the length of \\( \overline{j k} \\). round to the nearest whole number if necessary.

Explanation:

Step1: Use the tangent - secant theorem

The tangent - secant theorem states that if a tangent segment \(JK\) and a secant segment \(MK\) are drawn to a circle from an external point \(K\), then \(JK^{2}=KL\times KM\).
We know that \(KM = 12\) and \(KL=KM - ML\). Since \(ML = 10\), then \(KL=12 - 10=2\).

Step2: Calculate \(JK\)

Substitute \(KL = 2\) and \(KM = 12\) into the formula \(JK^{2}=KL\times KM\).
We get \(JK^{2}=2\times12\).
So \(JK^{2}=24\).
Take the square root of both sides: \(JK=\sqrt{24}\approx4.9\approx5\).

Answer:

\(5\)