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11.7
Chapter 11 Chapter 11 Parcc
Review
Test
Practice
12.1
12.2
12.3
12.4
12.5
12.6
PARCC
PARCC
PARCC
PARCC
No School
Homework
Section 12.1
Page 767-768
Exercises
6-20 all
23, 30
Class Rules!!!
1.
2.
3.
4.
5.
No getting up without asking
No talking until activities
Take Notes!
DO ALL EXAMPLES
Raise your hand
Tangent Lines
12.1
Objectives
• I can use Theorem 12.1 to calculate the
angles of a polygon
• I can use Theorem 12.2 to calculate the
side of a right triangle
• I can use Theorem 12.3 to calculate the
perimeter of a triangle that
circumscribes a circle
1 Geometry
Definitions
Circle-The set of all points
equidistant from a central point
B
Chord-A segment that connects
two points on a circle
A
C
Tangent of a circle-A line on the
same plane of a circle that
touches the circle at one point
Point B is called the Point of
Tangency
C12 Geometry
Theorem 12.1
Last Proof Type
• Proof by Contradiction
• Indirect Proof
2 Geometry
Example
Given: n 𝑖𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 ⊙ 𝑂 𝑎𝑡 𝑃
Prove: 𝑂𝑃 ⊥ 𝑛
Statements
n 𝑖𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 ⊙ 𝑂 𝑎𝑡 𝑃
Assume 𝑖𝑠 𝒏𝒐𝒕 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 ⊙ 𝑂 𝑎𝑡 𝑃
There must be a point L on n such that
𝑂𝐿 ⊥ 𝑛
Given
Proof by contradiction
Construction
There must be a point K on n such that
𝑃𝐿 ≅ 𝐾𝐿
Construction
∆𝑂𝑃𝐿 ≅ ∆𝑂𝐾𝐿
SAS
𝑂𝑃 ≅ 𝑂𝐾
CPCTC
𝑂𝑃 ⊥ 𝑛
For 𝑂𝑃 ≅ 𝑂𝐾, both P & K must be on ⊙ 𝑂
Reasons
Example
𝑀𝐿𝑎𝑛𝑑 𝑀𝑁 are tangents to the
circle. Calculate the value of x.
𝑚∠𝐿 𝑎𝑛𝑑 𝑚∠𝑁 = 90
𝑄𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑂𝐿𝑀𝑁
𝑥 = 360 − (117 + 90 + 90)
𝑥 = 63
3 Geometry
Example
𝐷𝐸 is tangent to the circle.
Calculate the value of x.
𝑚∠𝐷 = 90
𝑥 = 180 − (90 + 38)
𝑥 = 52
4 Geometry
Example
𝐴𝐷𝑎𝑛𝑑 𝐴𝐺 are tangents to the
circle. Calculate the value of x.
𝑚∠𝐷 𝑎𝑛𝑑 𝑚∠𝐺 = 90
𝑄𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑂𝐷𝐺𝐴
𝑥 = 360 − (152 + 90 + 90)
𝑥 = 28
Review
• I can use Theorem 12.1 to calculate the angles of a
polygon
• I can use Theorem 12.2 to calculate the side of a
right triangle
• I can use Theorem 12.3 to calculate the perimeter
of a triangle that circumscribes a circle
C12 Geometry
Theorem 12.2
5 Geometry
Example
𝐴𝐵 is tangent to the circle.
Calculate the value of x.
𝑎2 + 𝑏2 = 𝑐 2
𝑥 2 + 122 = (𝑥 + 8)2
𝑥 2 + 144 = 𝑥 2 + 16𝑥 + 64
144 = 16𝑥 + 64
80 = 16𝑥
𝑥=5
6 Geometry
Example
The triangle is tangent to the circle
at the right angle. Calculate the
value of x.
𝑎2 + 𝑏2 = 𝑐 2
𝑥 2 + 102 = (𝑥 + 6)2
𝑥 2 + 100 = 𝑥 2 + 12𝑥 + 36
100 = 12𝑥 + 36
64 = 12𝑥
16
𝑥=
3
7 Geometry
Example
Is 𝐿𝑀 a tangent to the circle?
𝑎2 + 𝑏2 = 𝑐 2
72 + 242 = 252
49 + 576 = 625
625 = 625
𝑌𝑒𝑠
Review
• I can use Theorem 12.1 to calculate the angles of a
polygon
• I can use Theorem 12.2 to calculate the side of a
right triangle
• I can use Theorem 12.3 to calculate the perimeter
of a triangle that circumscribes a circle
C12 Geometry
Theorem 12.3
8 Geometry
Example
Circle O inscribes ∆𝐴𝐵𝐶. Calculate
the perimeter of ∆𝐴𝐵𝐶 .
𝐴𝐷 = 10
𝐴𝐹 = 10
𝑃 = 66𝑐𝑚
𝐶𝐹 = 8
𝐶𝐸 = 8
𝐷𝐵 = 15
𝐸𝐵 = 15
8 Geometry
Example
Circle O inscribes ∆𝑃𝑄𝑅. The
perimeter of ∆𝑃𝑄𝑅 =88cm.
Calculate the length of 𝑄𝑋
𝑃𝑋 = 15
𝑃𝑍 = 15
𝑅𝑍 = 17
𝑅𝑌 = 17
𝑄𝑋 = 𝑥
𝑄𝑌 = 𝑥
88 = 15 + 15 + 17 + 17 + 𝑥 + 𝑥
88 = 64 + 2𝑥
𝑥 = 12
24 = 2𝑥
𝑄𝑋 = 12𝑐𝑚
Review
• I can use Theorem 12.1 to calculate the angles of a
polygon
• I can use Theorem 12.2 to calculate the side of a
right triangle
• I can use Theorem 12.3 to calculate the perimeter
of a triangle that circumscribes a circle
Homework
Section 12.1
Page 767-768
Exercises
6-20 all
23, 30
End of Class Rules!!!
1.
2.
3.
4.
5.
Clean Up your Area
Wait in your chairs
I dismiss not the bell
Push in your chairs
Keep it to a low roar