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