Question

determine if line ab is tangent to the circle.
1)
image of a circle with center, radius 12, line from center to a point on the circle (length 12), line from that point to b (length 8), and line from b to a (length 16), with a on the circle and ab connecting to a

Explanation:

Step1: Recall tangent - radius theorem

A tangent to a circle is perpendicular to the radius at the point of tangency. So, if \(AB\) is tangent to the circle at \(A\), then \(\angle OAB = 90^{\circ}\) (where \(O\) is the center of the circle), and triangle \(OAB\) should be a right - triangle. We can use the Pythagorean theorem to check. The Pythagorean theorem states that in a right - triangle with legs \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Here, the radius \(OA = 12\), \(AB = 16\), and the length from the external point \(B\) to the center \(O\) is \(OB=12 + 8=20\).

Step2: Apply Pythagorean theorem

Calculate \(OA^{2}+AB^{2}\):
\(OA^{2}=12^{2}=144\)
\(AB^{2}=16^{2}=256\)
\(OA^{2}+AB^{2}=144 + 256=400\)

Calculate \(OB^{2}\):
\(OB = 20\), so \(OB^{2}=20^{2}=400\)

Since \(OA^{2}+AB^{2}=OB^{2}\), by the converse of the Pythagorean theorem, triangle \(OAB\) is a right - triangle with \(\angle OAB = 90^{\circ}\). And since \(OA\) is the radius, \(AB\) is perpendicular to the radius at the point of tangency \(A\).

Answer:

Yes, line \(AB\) is tangent to the circle.