Question

given the circle below with secants (overline{hij}) and (overline{lkj}), find the length of (overline{hi}). round to the nearest tenth if necessary.

Explanation:

Step1: Use the secant - secant formula

If we have two secants \(HJ\) and \(LJ\) of a circle, then \(LJ\times KJ=HJ\times IJ\). Let \(HI = x\). Then \(HJ=x + 14\), \(LJ=14 + 17=31\), \(KJ = 17\), \(IJ = 14\).

Step2: Substitute into the formula

Substitute the values into the formula \(LJ\times KJ=HJ\times IJ\). We get \(31\times17=(x + 14)\times14\).

Step3: Expand the equation

First, calculate \(31\times17 = 527\) and \((x + 14)\times14=14x+196\). So the equation becomes \(527=14x + 196\).

Step4: Solve for \(x\)

Subtract \(196\) from both sides: \(527-196=14x\), \(331 = 14x\). Then \(x=\frac{331}{14}\approx19.0\).

Answer:

\(HI = 19.0\)