Question

given the circle below with secants \\( \overline { h i j } \\) and \\( \overline { l k j } \\). if \\( k j = 8, l k = 22 \\) and \\( i j = 10 \\), find the length of \\( \overline { h i } \\). round to the nearest tenth if necessary.

Explanation:

Step1: Apply the secant - secant theorem

The secant - secant theorem states that if two secants \( \overline{HIJ}\) and \( \overline{LKJ}\) are drawn to a circle from an external point \(J\), then \( (HI + IJ)\times IJ=(LK + KJ)\times KJ\).
Let \(HI = x\). We know that \(IJ = 10\), \(LK=22\), and \(KJ = 8\). Substituting these values into the formula, we get \((x + 10)\times10=(22 + 8)\times8\).

Step2: Expand and simplify the equation

First, expand the left - hand side: \(10x+100\).
Expand the right - hand side: \(30\times8 = 240\).
So the equation becomes \(10x+100 = 240\).

Step3: Solve for \(x\)

Subtract 100 from both sides of the equation: \(10x=240 - 100\).
\(10x=140\).
Divide both sides by 10: \(x=\frac{140}{10}=14\).

Answer:

\(14\)