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Tangents to Circles Geometry

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Tangents to Circles. Geometry. Objectives/Assignment. Identify segments and lines related to circles. Use properties of a tangent to a circle. Assignment: Chapter 10 Definitions Chapter 10 Postulates/Theorems pp. 599-601 #5-48 all. Some definitions you need. - PowerPoint PPT Presentation

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Page 1: Tangents to Circles

Tangents to Circles

Geometry

Page 2: Tangents to Circles

Objectives/Assignment

• Identify segments and lines related to circles.

• Use properties of a tangent to a circle.

• Assignment: – Chapter 10 Definitions– Chapter 10 Postulates/Theorems– pp. 599-601 #5-48 all

Page 3: Tangents to Circles

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.

Page 4: Tangents to Circles

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.

center

diameter

radius

Page 5: Tangents to Circles

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.

P

Q

R

S

Page 6: Tangents to Circles

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.

P

Q

R

S

Page 7: Tangents to Circles

k

j

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.

Page 8: Tangents to Circles

Ex. 1: 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.

a. AD

b. CD

c. EG

d. HB

J

H

B

A

CD

K

G

E

F

Page 9: Tangents to Circles

Ex. 1: 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.

a. AD – Diameter because it contains the center C.

b. CD

c. EG

d. HB

J

H

B

A

CD

K

G

E

F

Page 10: Tangents to Circles

Ex. 1: 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.

a. AD – Diameter because it contains the center C.

b. CD– radius because C is the center and D is a point on the circle.

J

H

B

A

CD

K

G

E

F

Page 11: Tangents to Circles

Ex. 1: 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.

c. EG – a tangent because it intersects the circle in one point.

J

H

B

A

CD

K

G

E

F

Page 12: Tangents to Circles

Ex. 1: 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.

c. EG – a tangent because it intersects the circle in one point.

d. HB is a chord because its endpoints are on the circle.

J

H

B

A

CD

K

G

E

F

Page 13: Tangents to Circles

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.

Page 14: Tangents to Circles

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 tangent

Page 15: Tangents to Circles

Concentric circles

• Circles that have a common center are called concentric circles.

Concentric circles

No points of intersection

Page 16: Tangents to Circles

Ex. 2: Identifying common tangents• Tell whether the

common tangents are internal or external.

j

k

C D

Page 17: Tangents to Circles

Ex. 2: Identifying common tangents• Tell whether the

common tangents are internal or external.

• The lines j and k intersect CD, so they are common internal tangents.

j

k

C D

Page 18: Tangents to Circles

Ex. 2: Identifying common tangents• Tell whether the

common tangents are internal or external.

• The lines m and n do not intersect AB, so they are common external tangents.

A

B

In a plane, the interior of a circle consists of the points that are inside the circle. The exterior of a circle consists of the points that are outside the circle.

Page 19: Tangents to Circles

14

12

10

8

6

4

2

5 10 15 20

BA

Ex. 3: Circles in Coordinate Geometry• Give the center

and the radius of each circle. Describe the intersection of the two circles and describe all common tangents.

Page 20: Tangents to Circles

14

12

10

8

6

4

2

5 10 15 20

BA

Ex. 3: Circles in Coordinate Geometry• Center of circle A is

(4, 4), and its radius is 4. The center of circle B is (5, 4) and its radius is 3. The two circles have one point of intersection (8, 4). The vertical line x = 8 is the only common tangent of the two circles.

Page 21: Tangents to Circles

Using properties of tangents

• The point at which a tangent line intersects the circle to which it is tangent is called the point of tangency. You will justify theorems in the exercises.

Page 22: Tangents to Circles

lQ

P

Theorem

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

• If l is tangent to Q at point P, then l

QP.⊥

l

Page 23: Tangents to Circles

lQ

P

Theorem (converse)

• In a plane, if a line is perpendicular to a radius of a circle at its endpoint on a circle, then the line is tangent to the circle.

• If l QP at P, then ⊥l is tangent to Q.

l

Page 24: Tangents to Circles

Ex. 4: Verifying a Tangent to a Circle• You can use the

Converse of the Pythagorean Theorem to tell whether EF is tangent to D.

• Because 112 _ 602 = 612, ∆DEF is a right triangle and DE is perpendicular to EF. So by Theorem 10.2; EF is tangent to D.

60

61

11

D

E

F

Page 25: Tangents to Circles

Ex. 5: Finding the radius of a circle• You are standing at

C, 8 feet away from a grain silo. The distance from you to a point of tangency is 16 feet. What is the radius of the silo?

• First draw it. Tangent BC is perpendicular to radius AB at B, so ∆ABC is a right triangle; so you can use the Pythagorean theorem to solve.

8 ft.

16 ft.

r

r

A

B

C

Page 26: Tangents to Circles

Solution: 8 ft.

16 ft.

r

r

A

B

C

(r + 8)2 = r2 + 162

Pythagorean Thm.

Substitute values

c2 = a2 + b2

r 2 + 16r + 64 = r2 + 256 Square of binomial

16r + 64 = 256

16r = 192

r = 12

Subtract r2 from each side.

Subtract 64 from each side.

Divide.

The radius of the silo is 12 feet.

Page 27: Tangents to Circles

Note:

• From a point in the circle’s exterior, you can draw exactly two different tangents to the circle. The following theorem tells you that the segments joining the external point to the two points of tangency are congruent.

Page 28: Tangents to Circles

Theorem • If two segments

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

• IF SR and ST are tangent to P, then SR ST.

P

T

S

R

Page 29: Tangents to Circles

Proof of Theorem 10.3

• Given: SR is tangent to P at R.

• Given: ST is tangent to P at T.

• Prove: SR ST

S P

T

R

Page 30: Tangents to Circles

S P

T

R

Proof

Statements:SR and ST are tangent to P

SR RP, STTPRP = TPRP TPPS PS

∆PRS ∆PTSSR ST

Reasons:Given

Tangent and radius are .

Definition of a circle

Definition of congruence.

Reflexive property

HL Congruence Theorem

CPCTC

Page 31: Tangents to Circles

Ex. 7: Using properties of tangents• AB is tangent to

C at B.• AD is tangent to

C at D.• Find the value of x.

11

AC

B

D

x2 + 2

Page 32: Tangents to Circles

Solution:11

AC

B

D

x2 + 2

11 = x2 + 2

Two tangent segments from the same point are

Substitute values

AB = AD

9 = x2Subtract 2 from each side.

3 = x Find the square root of 9.

The value of x is 3 or -3.

Page 33: Tangents to Circles

p = XY + YZ + ZW + WX Definition of perimeter p= XR + RY + YS + SZ + ZT + TW + WU + UX Segment Addition

Postulate= 11 + 8 + 8 + 6 + 6 + 7 + 7 + 11 Substitute.= 64 Simplify.

The perimeter is 64 ft.

XU = XR = 11 ft YS = YR = 8 ft ZS = ZT = 6 ftWU = WT = 7 ft

By Theorem 11-3, two segments tangent to a circle from a point outside the circle are congruent.

C is inscribed in quadrilateral XYZW. Find the perimeter of

XYZW.

.

Tangent Lines

Page 34: Tangents to Circles

Using Multiple Tangents

When a circle is inscribed in a triangle, the triangle is circumscribed about the circle.

What is the relationship between each side of the triangle and the circle?

Each segment is tangent to the circle, meaning each line is perpendicular to the radius forming a right angle.

Page 35: Tangents to Circles

Because opposite sides of a rectangle have the same measure, DW = 3 cm and OD = 15 cm.

Because OZ is a radius of O, OZ = 3 cm..

A belt fits tightly around two circular pulleys, as shown below.

Find the distance between the centers of the pulleys. Round

your answer to the nearest tenth.

Draw OP. Then draw OD parallel to ZW to form rectangle ODWZ, as shown below.

Real World and Tangent Lines

Page 36: Tangents to Circles

OD2 + PD2 = OP2 Pythagorean Theorem

152 + 42 = OP2 Substitute.

241 = OP2 Simplify.

The distance between the centers of the pulleys is about 15.5 cm.

OP 15.524175 Use a calculator to find the square root.

Because the radius of P is 7 cm, PD = 7 – 3 = 4 cm. .

Because ODP is the supplement of a right angle, ODP is also a right angle, and OPD is a right triangle.

(continued)