Find the measure of \(x\).
Identify the relationship between the tangent and the secant.
The tangent segment has length 4. The secant segment has external part \(x\) and internal part 5. The tangent segment and the secant segment intersect outside the circle.
Apply the tangent-secant theorem.
The tangent-secant theorem states that the square of the length of the tangent segment is equal to the product of the length of the external part of the secant segment and the length of the entire secant segment. Thus, \(4^2 = x(x+5)\).
Simplify the equation.
\(16 = x^2 + 5x\)
\(x^2 + 5x - 16 = 0\)
Solve the quadratic equation.
We can solve the quadratic equation using the quadratic formula:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
In our case, \(a=1\), \(b=5\), and \(c=-16\). Plugging these values into the formula gives us:
\(x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-16)}}{2(1)} = \frac{-5 \pm \sqrt{25 + 64}}{2} = \frac{-5 \pm \sqrt{89}}{2}\).
Since lengths must be positive, we have \(x = \frac{-5 + \sqrt{89}}{2}\).
\(\boxed{x = \frac{-5 + \sqrt{89}}{2}}\)
\(x = \frac{-5 + \sqrt{89}}{2}\)
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