Download Lines that intersect Circles

LINES THAT
INTERSECT CIRCLES
Geometry CP2 (Holt 12-1)
K. Santos
Circle definition
Circle: points in a plane that are a given distance (radius)
from a given point (center).
Circle P
Radius: center to a point on the circle
P
Interior & Exterior of a circle
Interior of a circle:
points inside the circle
Exterior of a circle:
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
• 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
Theorems 12-1-1 & 12-1-2
Radius perpendicular to tangent line (at point of tangency)
O
A
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—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
Example
Find x.
130°
x
Radius ⊥ tangents (right angles)
Sum of the angles in a quadrilateral are 360°
90 + 90 + 130 = 310
x = 360 - 310
x = 50°
Theorem 12-1-3
Same external point
tangents congruent
A
B
C
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