Prove that the perpendicular at the point of contact to the tangent to a circle passes through the center.

Solution:

The word "tangent" comes from the Latin word 'tangere,' which means "to touch."

A tangent line or tangent in geometry means a line or plane that intersects a curve or a curved surface at exactly one point.

Let O is the center of the given circle.

Draw a tangent LP touching the circle at point P.

Draw RP ⊥ LP at point P, such that point R lies on the circle.

∠OPL = 90° (radius ⊥ tangent)

Also, ∠RPL = 90° (Given)

∴ ∠OPL = ∠RPL

Now, this can only be possible when center O lies on the line RP.

Therefore, perpendicular at the point of contact to the tangent to a circle passes through the center of the circle.

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the center.

Summary:

The perpendicular at the point of contact to the tangent to a circle passes through the center is proved.