## Wednesday, August 1, 2007

### Geometry Chapter 10.1 Tangents to Circles

Chapter 10.1 Tangents to Circles

I) Vocabulary:
A) Circle - is the set of all pints in a plane that are equidistant from a given point, called the center of the circle.
B) Radius - the distance from the center to a point on the circle. Two circles are congruent if they have the same radius.
C) Diameter - the distance across the circle, through its center. The diameter is twice the radius.
D) Chord - is a segment whose endpoints are points on the circle.
E) Diameter - is a chord that passes through the center of the circle.
F) Secant - is a line that intersects a circle in two points.
G) Tangent - is a line in the plane of a circle that intersects the circle in exactly one point.
H) Tangent Circles - coplanar circles that intersect in one point.
I) Concentric - coplanar circles that have a common center.
J) Common tangents - a line or segment that is tangent to two coplanar circles.
1) Common internal tangent - intersects the segment that joins the centers of the two circles.
2) Common external tangent - does not intersect the segment that joins the centers of the two circles.
K) Interior of a circle - consists of the points that are inside the circle.
L) Exterior of a circle - consists of the points that are outside the circle.
M) Point of tangency - the point at which a tangent line intersects the circle to which it is tangent.
N) Theorems:
1) If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Example: Given line AB is tangent to circle C with point A on circle C, AC is a radius of 5 units, AB is 12 units, what is the length of line segment CB?
Since line AB is a tangent line, we know that AC and AB are perpendicular so angle CAB is a right angle so triangle CAB is a right triangle. Therefore we can use pythagorean theorem to find CB.
52 + 122 = (CB)2
25 + 144 = (CB)2
169 = (CB)2
13 = CB

2) In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
Example: Given radius DE = 11 units on circle D, there is an external point F, line segment DF = 61 and line segment EF = 60, is line EF a tangent line to circle D?
112 + 602 = 121 + 3600 = 3721
612 = 3721
Therefore, since the two smaller sides squared are equal to the longer side squared, this is the converse of pythagorean theorem so we know that angle DEF is a right angle, so therefore line EF is perpendicular to line segment DE so therefore EF is a tangent line to circle D.

3) If two segments from the same exterior point are tangent to a circle, then they are congruent.
Example: Given tangent SR = 2x + 7 and tangent SB = 5x - 8, where both tangents intersect at point S off of circle C with points R and B are on circle C, solve for x.
2x + 7 = 5x - 8
7 = 3x - 8
15 = 3x
5 = x