Transcript 9th Grade Geometry

9th 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
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
T
R
O
RT
Example: Find Lengths
ALGEBRA RS is tangent to
S
Find y.
20
P
Q
y
Q at point R.
16
R
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 Lengths
(SR)2 + (QR)2 = (SQ)2
162 + (QR)2 = 202
256 + (QR)2 = 400
(QR)2 = 144
QR = +12
Pythagorean Theorem
SR = 16, SQ = 20
Simplify
Subtract 256 from each side
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
CD is a tangent to
B at point D. Find a.
C
A. 15
B. 20
C. 10
D. 5
a
A
B
40
D
25
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
Determine whether BC is tangent to
A
C
7
9
7
A
7
B
First determine whether ABC is a right triangle
by using the converse of the Pythagorean
Theorem
Example: Identify Tangents
?
(AB)2 + (BC)2 = (AC)2
72
92
?
+ = 142
130 ≠ 196
Converse of the Pythagorean
Theorem
AB = 7, BC = 9, AC = 14
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 Tangents
Determine whether WE is tangent to
D.
E
16
24
10
D
10
W
First Determine whether
EWD is a right triangle by using the
converse of the Pythagorean Theorem
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.
Quick Review
Determine whether ED is a tangent to
D
A. Yes
√549
B. No
18
Q
C. Cannot be
determined
E
15
Q.
Quick Review
Determine whether XW is a tangent to
A. Yes
W
10
B. No
17
10
C. Cannot be
determined
V
10
X
V.
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
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
G
y-5
E
10
D
Example: Congruent Tangents
ED = FD
10 = y
Use the value of y to find x.
DG = DH
10 + (y - 5) = y + (x + 4)
10 + (10 - 5) = 10 + (x + 4)
15 = 14 + x
1=x
Answer: 1
Definition of congruent
segments
Substitution
Definition of congruent
segments
Substitution
y = 10
Simplify.
Subtract 14 from each
side
Quick Review
Find a. Assume that segments that appear
tangent to circles are tangent.
A. 6
30
B. 4
N
b
6 – 4a
C. 30
R
D. -6
A
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
J
16
M
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 Review
Triangle NOT is circumscribed about M. Find the
Perimeter of NOT if CT = NC – 28.
A. 86
B. 180
52
C
T
C. 172
D. 162
A
B
10
O
N