CK12 - Geometry

Proof.

We will provethis theoremby contradiction.Sincethe line is perpendicularto the radiusat its outerendpoint

it musttouchthe circleat point. For this line to be tangentto the circle,it mustonly touchthe circleat
this pointand no other.

Assumethat the line also intersectsthe circleat point.

Sinceboth and are radii of the circle, , and is isosceles.

This meansthat.

It is impossibleto havetwo right anglesin the sametriangle.

We arrivedat a contradictionso our assumptionmustbe incorrect.We concludethat line is tangent

to the circleat point.

Example 3

Determinewhether

is tangentto the circle.

is tangentto the circleif.

To showthat is a right trianglewe use the Converseof the PythagoreanTheorem: