chapter 10. a circle is the set of points in a plane that are equal distance, the radius (r), from a...
TRANSCRIPT
Chapter 10
A circle is the set of points in a plane that are equal distance, the radius (r), from a given point, the center, which is also in the plane.
Twice the radius of a circle is called the diameter of the circle.
r
The center of the circle is point A. A circle is named by its center. This circle is called circle A
Another definition of a radius of a circle is a segment whose endpoints are the center of the circle and a point on the circle. In circle A segment AB is a radius.
AB
Chord of a CircleA chord of a circle is a segment whose endpoints are on the circle.
A
B
CSegment CB is a chord of circle A.
A diameter of a circle is a segment that passes through the center of the circle and whose endpoints are on the circle.Segment DB is a diameter of circle A.
A
B
D
Tangent of a CircleA tangent of a circle is a line that intersects the circle in exactly one point.
A
B
ELine BE is a tangent of circle A.
Point B is the point of tangency.
Secant of a Circle
A secant of a circle is a line, a ray, or a segment that contains a chord of a circle.
A
BC
Line BC is a secant of circle A.
Tell whether the line or segment is best described as a radius, chord, diameter, secant, or tangent of circle C.
Example 1
a. radiusDC
b. secantBD�������������� �
c. diameterDE
d. tangentAE�������������� �
Use the diagram to find the given lengths
Example 2
a. Radius of 3 unitsB
b. Diameter of 4 unitsA
c. Radius of 2 unitsA
d. Diameter of 6 unitsB
Two coplanar circles can intersect in two points, one point, or no points.
Coplanar Circles
Coplanar circles that intersect in one point are called tangent circles.
Internally Tangent Circles
Externally Tangent Circles
Coplanar circles that have a common center are called concentric circles.
A line, ray, or segment that is tangent to two coplanar circles is called a common tangent.
Common Tangents
Tell how many common tangents the circles have and draw them.
Example 3
2 common tangents
4 common tangents
3 common tangents
1st Tangent Theorem
A line, a ray, or a segment is tangent to a circle if and only if it is in the same plane as the circle and is perpendicular to a radius of the circle at the point of intersection.
A
B
E
Line BE is a tangent of circle A if and only if it is perpendicular to radius AB at point B.
In the diagram, segment AB is a radius of circle A. Is segment BC tangent to circle A? Explain.
Segment BC is tangent to circle A if
segment BC radius AB at pt. B.
Example 4
267 44892 225 60 625 3600 4225
Therefore Δ ABC is not a right Δ and segmentBC is not perpendicular to radius AB.
In the diagram, S is a point of tangency. Find the radius r of circle T.
Example 5
ST SR
22 248 36r r 2 22304 72 1296r r r
1008 72r14 cmr
Tangent segments from a common external point are congruent.
2nd Tangent Theorem
A
BC
Given: tangent and tangent
intersecting at pt.
CA CB
C
Prove: AC BC
In circle C, segment DA is tangent at A and segment DB is tangent at B. Find x.
Example 6
AD BDAD BD
25 6 8x
33 6x5.5x