9th Grade Geometry

1
9th Grade Geometry

• Lesson 10-5 Tangents

2
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

3
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
4
Example Find Lengths

• ALGEBRA RS is tangent to Q at point R. Find y.

S
20
16
Q
R
P
y
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.
5
Example Find Lengths

• (SR)2 (QR)2 (SQ)2 Pythagorean
  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

6
Example

• CD is a tangent to B at point D. Find a.
• 15
• 20
• 10
• 5

C
a
B
D
A
40
25
7
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
8
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
9
Example Identify Tangents

• (AB)2 (BC)2 (AC)2 Converse of the
  Pythagorean Theorem
• 72 92 142 AB 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.

?
?
10
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
11
Example Identify Tangents

• (DW)2 (EW)2 (DE)2 Converse of the Pythagorean
  Theorem
• 102 242 262 DW 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.

?
?
12
Quick Review

• Determine whether ED is a tangent to Q.
• A. Yes
• B. No
• C. Cannot be
• determined

D
v549
18
Q
E
15
13
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
14
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
15
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
E
10
y - 5
16
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

17
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
18
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
19
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.

20
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