Download Lines that intersect Circles

LINES THAT
INTERSECT CIRCLES
Geometry 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
F
E
O
B
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
𝐴𝐵
• Tangent may be a line, ray, or segment
• The point where a circle and a tangent intersect is the
point of tangency
Point B
A
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
circles
externally tangent
circles
Common Tangent
Common tangent: a line that is tangent to two circles
Common external
tangents
common internal
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
Given: 𝐴𝐵 is tangent to circle O
Then: 𝐴𝐵⟘𝑂𝑃
P
B
Example
𝐸𝐷 is tangent to circle O. Radius is 5” and ED = 12”
Find the length of 𝑂𝐷 .
O
E
D
Remember Pythagorean theorem (let 𝑂𝐷 = x)
𝑥 2 = 52 + 122
𝑥 2 = 25 + 144
𝑥 2 = 169
x= 169
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
Given: 𝐴𝐵⟘𝑂𝑃
Then: 𝐴𝐵 is tangent to circle O
P
B
Example—Is there a tangent line?
Determine if there is a tangent line?
12
8
6
If there is a tangent then there must have been a right
angle (in a right triangle). Test for a right angle.
122 62 + 82
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: 𝐴𝐵 and 𝐶𝐵 are tangents to the circle
Then: 𝐴𝐵 ≅ 𝐶𝐵
Example:
R
𝑅𝑇 and 𝑅𝑆 are tangent to circle Q.
Find RS.
2n – 1
T
RT = RS
2n – 1 = n + 3
n–1=3
n=4
RS = n + 3
RS = 4 + 3
RS = 7
n+3
S