Geometry

The radius OP is perpendicular to the tangent PQ. Triangle OPQ is a right triangle with hypotenuse OQ = 12 and leg OP = 5. PQ = sqrt(OQ^2 - OP^2) = sqrt(144 - 25) = sqrt(119).

By definition, a tangent to a circle is always perpendicular to the radius at the point of contact.

By definition, a tangent to a circle touches the circle at exactly one point.

In quadrilateral OAPB, angles at A and B are 90 degrees. Angle APB = 80. Angle AOB = 180 - 80 = 100 degrees. Triangle OAB is isosceles with OA = OB. Angle OAB = (180 - 100) / 2 = 40 degrees.

A line that intersects a circle at two distinct points is defined as a secant.

By definition, a tangent line touches a circle at exactly one point.

A tangent to a circle forms a right-angled triangle with the radius and the line from the center to the external point. Using the Pythagorean theorem, the square of the tangent length plus the square of the radius equals the square of the distance from the center: t^2 + 8^2 = 10^2. Thus, t^2 = 100 - 64 = 36, so t = 6cm.

By definition, the point where a tangent touches a circle is called the point of contact.

A line intersecting a circle at two distinct points is defined as a secant line.

By definition, a line drawn perpendicular to the radius at its point of contact with the circle is a tangent.

In a circle, the diameter perpendicular to a chord bisects the chord. If the chord is parallel to the tangent at point A, and AB is the diameter through A, the chord is perpendicular to the diameter AB. Therefore, the diameter bisects the chord.

In a right triangle ABC with angle B = 90, let AB be the diameter of a circle. The circle intersects AC at P. By property of circles, angle APB = 90, so BP is the altitude to the hypotenuse. The tangent at P to the circle is perpendicular to the radius OP. Geometric analysis confirms this tangent bisects BC.