Question

determine if line ab is tangent to the circle

Explanation:

Step1: Calculate the length from the center to point B

The length from the center to point B is \(4 + 6=10\).

Step2: Apply the Pythagorean theorem

If \(AB\) is tangent to the circle, then \(\triangle ABA'\) (where \(A'\) is the center) is a right - triangle. According to the Pythagorean theorem, in a right - triangle, \(a^{2}+b^{2}=c^{2}\). Here, if \(AB = 8\), the radius \(r = 6\), and the distance from the center to \(B\) is \(d=10\), we check \(8^{2}+6^{2}\) and \(10^{2}\).

$$8^{2}+6^{2}=64 + 36=100$$

$$10^{2}=100$$

Answer:

Since \(8^{2}+6^{2}=10^{2}\), line \(AB\) is tangent to the circle.