LINES THAT INTERSECT CIRCLESGeometry CP1 (Holt 12-1) K. Santos
Circle definition
Circle: set of all points in a plane that are a given distance (radius) from a given point (center).
Circle P P
Radius: is a segment that connects the center of the circle
to a point on the circle
Interior & Exterior of a circle
Interior of a circle:
set of all the points inside the circle
Exterior of a circle:
set of all points outside the circle
exterior interior
Lines & Segments that intersect a circle
A
G O B
F
E C
D
Chord: is a segment whose endpoints
lie on a circle.
Diameter: -a chord that contains the center
-connects two points on the circle and passes through the center
Secant: line that intersects a circle at two points
Tangent• A tangent to a circle is a line in the plane of the circle that
intersects the circle in exactly one point
• Tangent may be a line, ray, or segment
• The point where a circle and a tangent intersect is the point of tangency A
Point B
B
Pairs of circles
Congruent Circles: two circles that have congruent radii
Concentric Circles: coplanar circles with the same center
Tangent Circles
Tangent Circles: coplanar circles that intersect at exactly one point
Internally tangent externally tangent
circles circles
Common Tangent
Common tangent: a line that is tangent to two circles
Common external common internal
tangents tangents
Theorem 12-1-1
If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
O
A P B
Given: is tangent to circle O
Then:
Example
is tangent to circle O. Radius is 5” and ED = 12”
Find the length of .
O
E D
Remember Pythagorean theorem (let = x)
= +
= 25 + 144
= 169
x=
x = 13
Example
Find x.
130 x
Radius perpendicular to tangents (right angles)
Sum of the angles in a quadrilateral are 360
90 + 90 + 130 + x = 360
310 + x = 360
x = 50
Theorem 12-1-2
If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.
O
A P B
Given:
Then: is tangent to circle O
Example—Is there a tangent line?
Determine if there is a tangent line?
12 6
8
If there is a tangent then there must have been a right angle (in a right triangle). Test for a right angle.
+
144 36 + 64
144 100 so there is no right angle, no tangent line
Theorem 12-1-3
If two segments are tangent to a circle from the same point,
then the segments are congruent.
A
B
C
Given: and are tangents to the circle
Then:
Example: R
and are tangent to circle Q. 2n – 1 n + 3
Find RS.
T S
RT = RS
2n – 1 = n + 3
n – 1 = 3
n = 4
RS = n + 3
RS = 4 + 3
RS = 7