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Geometry - PowerPoint Presentation

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Geometry - PPT Presentation

95 Tangents to Circles Objectives Identify segments and lines related to circles Use properties of a tangent to a circle Some definitions you need Circle set of all points in a plane that are ID: 419213

tangent circle center circles circle tangent circles center radius called point points common line segment diameter definitions tangents concentric segments find secant

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Slide1

Geometry

9.5

Tangents to CirclesSlide2

Objectives

Identify segments and lines related to circles.

Use properties of a tangent to a circle

.Slide3

Some definitions you need

Circle

– set of all points in a plane that are

equidistant

from a given point called a

center of the circle. A circle with center P is called “circle P”, or P.The distance from the center to a point on the circle is called the radius of the circle. Two circles are congruent if they have the same radius.Slide4

Some definitions you need

The distance across the circle, through its center is the

diameter

of the circle. The diameter is twice the radius.

The terms

radius and diameter describe segments as well as measures.Slide5

Some definitions you need

A radius is a segment whose endpoints are the center of the circle and a point on the circle

.

QP, QR, and QS are radii of Q. All radii of a circle are congruent.Slide6

Some definitions you need

A

chord

is a segment whose endpoints are points on the circle. PS and PR are

chords

.A diameter is a chord that passes through the center of the circle. PR is a diameter.Slide7

Some definitions you need

A

secant

is a line that intersects a circle in two points.

Line

k is a secant.A

tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line

j is a tangent. Slide8

Identifying

Special Segments and Lines

Tell

whether the line

or segment

is best described as a chord, a secant, a tangent, a diameter, or a radius of C.ADCDEG

HBSlide9

More information you need--

In a plane, two circles can intersect in

two points, one point, or no points.

Coplanar circles that intersect in one point are called

tangent circles

. Coplanar circles that have a common center are called concentric.

2 points of intersection.Slide10

Tangent circles

A line or segment that is tangent to two coplanar circles is called a common tangent.

A common internal tangent intersects the segment that joins the centers of the two circles.

A

common external tangent does not intersect the segment that joins the center of the two circles.

Internally tangent

Externally tangentSlide11

Concentric circles

Circles that have a common center are called concentric circles.

Concentric circles

No points of intersectionSlide12

Identifying

common tangents

Tell whether the common tangents are internal or external. Slide13
Slide14
Slide15

Verifying

a Tangent to a Circle

Determine if EF

is tangent to

circle D.

Because 112 + 602 = 612

DEF is a right triangle and DE is perpendicular to EF.

EF

is tangent to

circle D

.Slide16

Find Radius

(r + 8)

2

= r

2

+ 16

2

Pythagorean Thm.

Substitute values

c

2

= a

2

+ b

2

r

2

+ 16r + 64 = r

2

+ 256

Square of binomial

16r + 64 = 256

16r = 192

r = 12

Subtract r

2

from each side.

Subtract 64 from each side.

Divide.

The radius of the

circle

is 12 feet.Slide17

Using properties of tangents

AB is tangent to C at B.

AD is tangent

to

C at D.

Find the value of x.

x

2

+ 2Slide18

x

2

+ 2

11 = x

2

+ 2

Two tangent segments from the same point are

Substitute values

AB = AD

9 = x

2

Subtract 2 from each side.

3 = x

Find the square root of 9.

The value of x is 3 or -3.

AB is tangent to C at B.

AD is tangent

to

C at D.

Find the value of x.

x

2

+ 2Slide19

PracticeSlide20

X ²= 400+ 225

x

² = 625

X = 25Slide21

40² + 30

² = 2500

Hypotenuse = √2500 = 50

X

= 50-30 =

20

Hypotenuse = 21 + 8 = 29

29

² - 21 ² = 400

X

=

√400 =

20Slide22

AC = BC

40 = 3x + 4

36 = 3x

X = 9

Pythagorean Formula

BP² = AB

² + AP²

(8 + x)

² = x ² + 12²

x

² + 16x + 64 = x² + 144

16x = 80

X = 5Slide23

CD = ED = 4

BC = AB = 3

X = BC + CD

X

= 3+4 =

7

Trigonometric Formula

Cos 30 = 6 / x x = 6 / Cos 30 x = 6.9

30°Slide24

Pythagorean Formula

BP² = CP

² + CB²

X

² = 12 ² - 4²

x² = 144-16X = 11.3

PB = 6

PC = 6 + 4 = 10

PQ = 6

Pythagorean Formula

QC

² = PC ² -PQ ²

QC

²

= 100-36 = 64

QC = 8

Pythagorean Formula

CD

² = CQ ² - QD ²

CD

² = 64 – 40.96

CD = 4.8