Answer: So the vertex of the angle is located at the center of the circle, which is 8 centimeters away from the nearest centimeter.
Explanation:
Let O be the center of the circle, and let A and B be the points of tangency of the two tangents with the circle. Since OA and OB are radii of the circle, they have the same length of 8 centimeters.
Let C be the vertex of the angle formed by the two tangents. Since the tangents are perpendicular to the radii at the points of tangency, we have that angle AOC = angle BOC = 90 degrees.
Since the measure of the angle formed by the two tangents is 80 degrees, we have that angle AOB = 180 - 80 - 80 = 20 degrees.
Let D be the foot of the perpendicular from C to line AB. Then angle OCD = 90 - 20/2 = 80 degrees, so triangle OCD is an isosceles triangle. Therefore, we have that OD = OC = 8 centimeters.
Finally, since triangle OCD is a right triangle, we can use the Pythagorean theorem to find the length of CD. We have:
CD^2 = OD^2 - OC^2 = 8^2 - 8^2 = 0
Therefore, CD = 0 centimeters.
So the vertex of the angle is located at the center of the circle, which is 8 centimeters away from the nearest centimeter.