How do I prove that a line perpendicular to a radius at its outer endpoint is tangent to a circle?

1.) Line L is in the plane of Circle Q; Line L is perpendicular to radius QR at R            Given
2.)  definition of perpendicular lines
3.) definition of right angle
3.)Line L is tangent to Circle Q        definition of tangent

its kind of a given by your definition but wikipedia defines a tangent line to intersect that curve only at one point so if you draw a circle in the first quadrant of a graph.  (west side of circle touches y axis, south side of circle touches x axis) you could show that the y axis and x axis are indeed perpendicular to that circles radius and that they touch the curve of the circle in only one point because all other x and y components of the circles circumference points are greater than 0 but i believe this method would require calculus to be used.

suppose you have a secant line, then what is the point on this secant line which is closest to the center? and what angle does a line from the center to this point make with the secant line itself?

now by the definition of the tangent line, each point on the tangent line except for one has a distance from the center of the circle greater than r. so that tangent point is the closest point to the center. but the tangent line is a secant line with the two points of intersection the same, so what must that there angle be?

If a line is perpendicular to the radius of a circle at its outer endpoint, then the line is tangent to the circle.

Proof: We will prove this theorem by contradiction.
Since line BC is perpendicular to the radius at its outer endpoint it must touch the circle at point B. For this line to be tangent to the circle, it must only touch the circle at this point and no other.

Assume that line BC also intersects the circle at point C.

Since both AB and AC are radii of the circle, , and  is isosceles.
This means that .
It is impossible to have two right angles inside a triangle.

Since we arrived at a falsehood, our assumption must be incorrect.
We conclude that line BC is tangent to the circle at point B.