99thth Grade Geometry Grade Geometry

Lesson 10-5: TangentsLesson 10-5: Tangents

Main IdeaMain Idea

Use properties of tangents!Use properties of tangents!

Solve problems involving circumscribed Solve problems involving circumscribed polygonspolygons

New VocabularyNew VocabularyTangent Tangent – Any line that touches a curve in exactly one placeAny line that touches a curve in exactly one place

Point of TangencyPoint of Tangency– The point where the curve and the line meetThe point where the curve and the line meet

Theorem 10.9Theorem 10.9

If a line is tangent to a circle, then it is If a line is tangent to a circle, then it is perpendicular to the radius drawn to the perpendicular to the radius drawn to the point of tangency.point of tangency.– Example: If RT is a tangent, OR RTExample: If RT is a tangent, OR RT

T

R

O

Example: Find LengthsExample: Find Lengths

ALGEBRAALGEBRA RSRS is tangent to is tangent to QQ at point at point RR. . Find Find yy..

y

20 16

S

RPQ

Because the radius is perpendicular to the tangent at the point of tangency, QR SR. This makes SRQ a right angle and SRQ a right triangle. Use the Pythagorean Theorem to find QR, which is one-half the length y.

Example: Find LengthsExample: Find Lengths

((SR)SR)2 2 + (QR)2 = (SQ)2 Pythagorean Theorem

16162 + (QR) + (QR)2 = 20 = 202 SR = 16, SQ = 20

256 + (QR)2 = 400 Simplify

(QR)2 = 144 Subtract 256 from each side

QR = +12 Take the square root of each side

Because y is the length of the diameter, ignore the negative result. Thus, y is twice QR or y = 2(12) = 24

Answer: y = 24

ExampleExample

CD CD is a tangent to is a tangent to BB at point at point D.D. Find Find a.a.

A.A. 1515

B. 20

C. 10

D. 5

40

25

a

C

DAB

Theorem 10.10Theorem 10.10

If a line is perpendicular to a radius of a If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the circle at its endpoint on the circle, then the line is tangent to the circle.line is tangent to the circle.– Example: If Example: If OR RT, RTOR RT, RT is a tangent. is a tangent.

R

T

O

Example: Identify TangentsExample: Identify Tangents

Determine whether Determine whether BCBC is tangent to A is tangent to A

7

79

C

BA

7

First determine whether ABC is a right triangle by using the converse of the Pythagorean Theorem

Example: Identify TangentsExample: Identify Tangents

((ABAB))22 + ( + (BCBC))22 = ( = (ACAC))22 Converse of the Pythagorean Converse of the Pythagorean Theorem Theorem

7722 + 9 + 922 = 14 = 1422 ABAB = 7, = 7, BCBC = 9, = 9, ACAC = 14 = 14

130 130 ≠ 196≠ 196 SimplifySimplify

Because the converse of the Pythagorean Theorem did Because the converse of the Pythagorean Theorem did not prove true in this case, not prove true in this case, ABCABC is not a right triangle is not a right triangle

Answer:Answer: So, So, BCBC is not a tangent to is not a tangent to AA..

?

?

Example: Identify TangentsExample: Identify Tangents

Determine whether Determine whether WEWE is tangent to is tangent to DD..

10

1024

E

WD

16

First Determine whether EWD is a right triangle by using the converse of the Pythagorean Theorem

Example: Identify TangentsExample: Identify Tangents

((DWDW))22 + ( + (EWEW))22 = ( = (DEDE))22 Converse of the Converse of the Pythagorean Pythagorean Theorem Theorem

101022 +24 +2422 = 26 = 2622 DWDW = 10, = 10, EWEW = 24, = 24, DEDE = 26 = 26

676 = 676676 = 676 Simplify.Simplify.

Because the converse of the Pythagorean Theorem is Because the converse of the Pythagorean Theorem is true, true, EWDEWD is a right triangle and is a right triangle and EWDEWD is a right is a right angle.angle.

Answer:Answer: Thus, Thus, DW DW WEWE, making , making WEWE a tangent to a tangent to DD..

?

?

Quick ReviewQuick Review

Determine whether Determine whether EDED is a tangent to is a tangent to QQ..

A. YesA. Yes

B. NoB. No

C. Cannot be C. Cannot be

determineddetermined

15

√549

18

D

EQ

Quick ReviewQuick Review

Determine whether Determine whether XWXW is a tangent to is a tangent to VV..

A. YesA. Yes

B. NoB. No

C. Cannot be C. Cannot be

determineddetermined

10

1017

W

XV

10

Theorem 10.11Theorem 10.11

If two segments from the same exterior If two segments from the same exterior point are tangent to a circle, then they are point are tangent to a circle, then they are congruentcongruent– Example: Example: AB AB ≈ AC≈ AC

B

C A

Example: Congruent TangentsExample: Congruent Tangents

ALGEBRAALGEBRA Find Find x. x. Assume that segments Assume that segments that appear tangent to circles are tangent.that appear tangent to circles are tangent.

H

G D

ED and FD are drawn from the same exterior point and are tangent to S, so ED ≈ FD. DG and DH are drawn from the same exterior point and are tangent to T, so DG ≈ DH

F

E 10

y

x + 4

y - 5

Example: Congruent TangentsExample: Congruent Tangents

EDED = = FDFD Definition of congruent Definition of congruent segmentssegments10 = 10 = yy SubstitutionSubstitution

Use the value of Use the value of yy to find to find xx.. DGDG = = DHDH Definition of congruent Definition of congruent

segmentssegments10 + (10 + (yy - 5) = - 5) = yy + ( + (xx + 4) + 4) SubstitutionSubstitution10 + (10 - 5) = 10 + (10 + (10 - 5) = 10 + (xx + 4) + 4) yy = 10 = 1015 = 14 + 15 = 14 + x x Simplify.Simplify.1 = 1 = xx Subtract 14 from each Subtract 14 from each

sidesideAnswer:Answer: 11

Quick ReviewQuick Review

Find a. Assume that segments that appear Find a. Assume that segments that appear tangent to circles are tangent.tangent to circles are tangent.

A.A. 66

B.B. 44

C.C. 3030

D.D. -6-6 R

A

N6 – 4a

b

30

Example: Triangles Circumscribed Example: Triangles Circumscribed About a CircleAbout a Circle

Triangle Triangle HJKHJK is circumscribed about is circumscribed about GG. . Find the perimeter of Find the perimeter of HJKHJK if if NKNK = = JLJL +29 +29

L

N

M

K

H

J 16

18

Example: Triangles Circumscribed Example: Triangles Circumscribed About a CircleAbout a Circle

Use Theorem 10.11 to determine the equal measures:Use Theorem 10.11 to determine the equal measures:

JMJM = = JLJL = 16, = 16, JHJH = = HNHN = 18, and = 18, and NKNK = = MKMK

We are given that We are given that NKNK = = JLJL + 29, so + 29, so NKNK = 16 + 29 or 45 = 16 + 29 or 45

Then Then MKMK = 45 = 45

PP = = JMJM + + MKMK + + HNHN + + NKNK + + JLJL + + LHLH Definition of Definition of perimeterperimeter

= 16 + 45 + 18 + 45 + 16 + 18 or 158= 16 + 45 + 18 + 45 + 16 + 18 or 158 Substitution Substitution

Answer:Answer: The perimeter of The perimeter of HJKHJK is 158 units. is 158 units.

Quick ReviewQuick Review

Triangle NOT is circumscribed about M. Find the Triangle NOT is circumscribed about M. Find the Perimeter of NOT if CT = NC – 28.Perimeter of NOT if CT = NC – 28.

A.A. 8686

B.B. 180180

C.C. 172172

D.D. 162162

O

N

A

T

B

C52

10