Tangents To recognize tangents and use the properties

Tangents To recognize tangents and use the properties of tangents

Definition Tangent – A line that intersects a circle in exactly 1 pt. Pt of tangency – Pt where a tangent line intersects a circle Secant – A line that intersects a circle in 2 pts A circle separates a plane into 3 parts Secant circle Tangent line interior exterior Pt of tangency

Theorem If a line is tangent to a circle , then it is perpendicular to the radius drawn to the pt of tangency Converse – If a line is perpendicular to a radius of a circle at the end pt on the circle, then the line is a tangent of the circle

Example ALGEBRA is tangent to Answer: Thus, y is twice at point R. Find y. .

Example Is AB tangent to circle C? B 4 2 ST is tangent to o. Q. Find r A 5 C No 22 + 4 2 = 5 2 18 T 24 r Q r S 242 + r 2 = (18 + r)2 576 + r 2 = 324 + 36 r + r 2 576 = 324 + 36 r 252 = 36 r 7=r

Definition: Common tangent – a line or line segment that is tangent to 2 circles in the same plane There are 2 types of common tangents Common external tangents Tangents do not intersect the segment connecting the centers of the circle Common internal tangents Tangents intersect the segment connecting the centers

Theorem If 2 segments from the same exterior pt are tangent to a circle, then they are congruent

Example ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent. ED congruent FD…so y = 10 EG is congruent to FH…so y-5= x+4 10 – 5 = x + 4 5= x+4 1=x

Example: Find C 2 c 2 +9 c+6 4 1 + 9 c 2 c 2 + 9 c + 6 = 9 c + 14 2 c 2 + 6 = 14 2 c 2 = 8 c 2 = 4 c = 2 and -2 Can’t be -2 because that will make the segment – in length

Circumscribed Polygons A polygon is circumscribed about a circle, if each side of the polygon is tangent to the circle

Example Triangle HJK is circumscribed about perimeter of HJK if Find the 16 16 +2 9 18 16 + 29 Answer: 158 units

Homework Put this in your agenda Pg 655 1 – 10, 15 – 26