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9 th Grade Geometry. Lesson 10-5: Tangents. Main Idea . Use properties of tangents! Solve problems involving circumscribed polygons. New Vocabulary. Tangent Any line that touches a curve in exactly one place Point of Tangency The point where the curve and the line meet. Theorem 10.9.

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9 th grade geometry

9th Grade Geometry

Lesson 10-5: Tangents


Main idea
Main Idea

  • Use properties of tangents!

  • Solve problems involving circumscribed polygons

New Vocabulary

  • Tangent

    • Any line that touches a curve in exactly one place

  • Point of Tangency

    • The point where the curve and the line meet


Theorem 10 9
Theorem 10.9

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

    • Example: If RT is a tangent, OR RT

T

R

O


Example find lengths
Example: Find Lengths

ALGEBRARS is tangent to Q at point R. Find y.

S

20

16

Q

P

R

y

Because the radius is perpendicular to the tangent at the point of tangency, QRSR. 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 lengths1
Example: Find Lengths

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

162 + (QR)2 = 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


Example
Example

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

  • 15

  • 20

  • 10

  • 5

C

a

B

A

D

40

25


Theorem 10 10
Theorem 10.10

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

    • Example: If OR RT, RT is a tangent.

R

T

O


Example identify tangents
Example: Identify Tangents

Determine whether BC is tangent to A

C

7

9

7

A

B

7

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


Example identify tangents1
Example: Identify Tangents

(AB)2 + (BC)2 = (AC)2 Converse of the Pythagorean Theorem

72 + 92 = 142AB = 7, BC = 9, AC = 14

130 ≠ 196 Simplify

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

Answer:So, BC is not a tangent to A.

?

?


Example identify tangents2
Example: Identify Tangents

Determine whether WE is tangent to D.

E

16

24

10

D

W

10

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


Example identify tangents3
Example: Identify Tangents

(DW)2 + (EW)2 = (DE)2 Converse of the Pythagorean Theorem

102 +242 = 262DW = 10, EW = 24, DE = 26

676 = 676 Simplify.

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

Answer:Thus, DW WE, making WE a tangent to D.

?

?


Quick review
Quick Review

Determine whether ED is a tangent to Q.

A. Yes

B. No

C. Cannot be

determined

D

√549

18

Q

E

15


Quick review1
Quick Review

Determine whether XW is a tangent to V.

A. Yes

B. No

C. Cannot be

determined

W

10

17

10

V

X

10


Theorem 10 11
Theorem 10.11

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

    • Example: AB ≈ AC

B

C

A


Example congruent tangents
Example: Congruent Tangents

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

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

H

x + 4

F

y

D

G

y - 5

E

10


Example congruent tangents1
Example: Congruent Tangents

ED = FD Definition of congruent segments

10 = y Substitution

Use the value of y to find x.

DG = DH Definition of congruent segments

10 + (y - 5) = y + (x + 4) Substitution

10 + (10 - 5) = 10 + (x + 4) y = 10

15 = 14 + x Simplify.

1 = x Subtract 14 from each side

Answer:1


Quick review2
Quick Review

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

  • 6

  • 4

  • 30

  • -6

30

N

b

6 – 4a

R

A


Example triangles circumscribed about a circle
Example: Triangles Circumscribed About a Circle

Triangle HJK is circumscribed about G. Find the perimeter of HJK if NK = JL +29

H

N

18

K

L

M

16

J


Example triangles circumscribed about a circle1
Example: Triangles Circumscribed About a Circle

Use Theorem 10.11 to determine the equal measures:

JM = JL = 16, JH = HN = 18, and NK = MK

We are given that NK = JL + 29, so NK = 16 + 29 or 45

Then MK = 45

P = JM + MK + HN + NK + JL + LH Definition of

perimeter

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

Answer:The perimeter of HJK is 158 units.


Quick review3
Quick Review

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

  • 86

  • 180

  • 172

  • 162

N

52

C

T

A

B

10

O


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