1. 8.4 Tangent Properties
A tangent line is a line that meets the circle at exactly one point. The point of tangency.

2. Tangent Property
If a line is tangent to a circle, then the radius drawn to the point of tangency will be perpendicular to the tangent
OP AB
∟OPA = 90°
∟OPA is a right angle O A
P (Point of Tangency) B

3. AB OA by the tangent property Therefore ΔOAB is a right triangle
Example 1: Given AB = 12, OB = 13 and A is the point of tangency Find Radius of the circle
Solution 1:
AB OA by the tangent property
Therefore ΔOAB is a right triangle
By Pythagorean Theorem:
(OB)2 = (OA)2 + (AB)2
132 = (OA)
169 – 144 = (OA)2
25 = (OA)2
OA = 5
Therefore the RADIUS = 5 A O B

4. Tangents from an External Point
It is possible to draw to tangent from a point exterior to the circle. A P O B

5. Example 2: Given: ∟O = 130° and A, B are points of tangency Find: ∟P
Solution 2:
∟A = 90° and ∟B = 90° by the tangent property
AOBP is a quadrilateral so ∟O + ∟A + ∟P + ∟B = 360°
130° + 90° + ∟P + 90° = 360°
∟P = 50° A O P B