Lesson 6.1 Tangents to Circles

Goal 1 Communicating About Circles

Goal 2 Using Properties of Tangents

Communicating About Circles

Circle Terminology:A CIRCLE is the set of all points in the plane that are a given distance from a given point.  The given point is called the CENTER of the circle.

A

A circle is named by its center point.

“Circle A”

or A

Communicating About Circles

RadiusChord

Diameter

Tangent

Secant

Parts of a CirclePoint of Tangency

Communicating About Circles

A C

A C

A C

In a plane, two circles can intersect in two points, one point, or no points.

Two PointsOne Point

No Point

Coplanar Circles that intersect in one point are called Tangent Circles

Communicating About Circles

Tangent Circles

A line tangent to two coplanar circles is called a Common Tangent

Communicating About Circles

Concentric CirclesTwo or more coplanar circles that share the same center.

Communicating About Circles

Common Internal Tangents

Common External Tangents

Common External Tangent does not intersect the segment joining the centers of the two circles.

Common Internal Tangent intersects the segment joining the centers of the two circles.

Communicating About Circles

A circle divides a plane into three parts

1.Interior

2.Exterior

3.On the circle

Using Properties of Tangents

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

Radius to a Tangent Conjecture

The Converse then states:  In a plane, if a line is perpendicular to a radius of a circle at the endpoint on the circle, then the line is a tangent of the circle.

Using Properties of Tangents

45

43

11

R

S T

Is TS tangent to R? Explain

If the Pythagorean Theorem works then the triangle is a right triangle TS is tangent

222 114345 ?

12118492025 ?

19702025 NO!

Using Properties of Tangents

You are standing 14 feet from a water tower. The distance from you to a point of tangency on the tower is 28 feet. What is the radius of the water tower?

r

14 ft

28 ft

r

R

S T

Radius = 21 feet Tower

222 2814 rr

222 2819628 rrr

78419628 r58828 r

21r

Using Properties of Tangents

If two segments from the same exterior point are tangent to the circle, then they are congruent.

Tangent Segments Conjecture

ACAB

Using Properties of Tangents

21

R

S

U

V

x2 - 4US is tangent to R at S. is tangent to R at V.

Find the value of x.

UV

2142 x

252 x5x

Using Properties of Tangents

x

z 15

y36

R

S

U

V

Find the values of x, y, and z.All radii are =

y = 15

222 1536 x22512962 x

15212 x39x

Tangent segments are =

z = 36