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

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

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)2 + 122 169 – 144 = (OA)2 25 = (OA)2 OA = 5 Therefore the RADIUS = 5 A O B

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

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