Question

1. in the accompanying figure, \\( \overrightarrow { a d } \\) is tangent to circle \\( o \\) at \\( d \\) and \\( \overline { a b c } \\) is a secant. if \\( a d = 4 \\) and \\( a c = 8 \\), find \\( a b \\).

Explanation:

Step1: Use the tangent - secant theorem

The tangent - secant theorem states that if a tangent segment \(AD\) and a secant segment \(ABC\) are drawn to a circle from an external point \(A\), then \(AD^{2}=AB\times AC\).
Let \(AB = x\). We know that \(AD = 4\) and \(AC=8\). Substituting these values into the formula, we get \(4^{2}=x\times8\).

Step2: Solve the equation for \(x\)

The equation \(16 = 8x\) can be solved by dividing both sides of the equation by \(8\). So, \(x=\frac{16}{8}\).

Answer:

\(AB = 2\)