Question

given the circle below with secants \\( \overline { h i j } \\) and \\( \overline { l k j } \\), find the length of \\( 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 \(HJ\) and \(LJ\) are drawn to a circle from an external point \(J\), then \(LJ\times KJ=HJ\times IJ\). Let \(HI = x\), then \(HJ=x + 14\), \(LJ=14 + 17=31\), and \(KJ = 17\), \(IJ = 14\).
Substitute into the formula: \(31\times17=(x + 14)\times14\).

Step2: Expand and solve the equation

First, expand the right - hand side: \(31\times17 = 14x+14\times14\).
Calculate \(31\times17 = 527\) and \(14\times14 = 196\). So the equation becomes \(527=14x + 196\).
Subtract 196 from both sides: \(14x=527 - 196\).
\(14x=331\).
Then \(x=\frac{331}{14}\approx23.6\).

Answer:

\(23.6\)