1. Tangents and Circles
Topic Index | Geometry Index | Regents Exam Prep Center
A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one
point.
If you spin an object in a circular
orbit and release it, it will travel on
a path that is tangent to the circular
orbit.
Theorem: If a line is tangent to a circle, it is perpendicular to the radius drawn to the
point of tangency.
Tangent segments to a circle from the same external point are congruent.
Theorem:
(You may think of this as the "Hat" Theorem because the diagram looks
like a circle wearing a pointed hat.)

2. This theorem can be proven using congruent triangles and the previous theorem. The triangles shown below
are congruent by the Hypotenuse Leg Postulate for Right Triangles. The radii (legs) are congruent and the
hypotenuse is shared by both triangles. By using Corresponding Parts of Congruent Triangles are
Congruent, this theorem is proven true.
Common tangents are lines or segments that are tangent to more than
Common Tangents:
one circle at the same time.
4 Common Tangents 3 Common Tangents 2 Common Tangents
(2 completely separate circles) (2 externally tangent circles) (2 overlapping circles)
2 external tangents (blue)
2 external tangents (blue) 2 external tangents (blue) 0 internal tangents
2 internal tangents (black) 1 internal tangent (black)
1 Common Tangent 0 Common Tangents
(2 internally tangent circles) (2 concentric circles) (one circle floating inside the other,
Concentric circles are circles without touching)
with the same center.
1 external tangent (blue)
0 internal tangents 0 external tangents 0 external tangents
0 internal tangents 0 internal tangents