unit 3 lesson 5 tangents, arcs, and...
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Unit 3Lesson 5
Tangents, arcs, and sectors
What are we going to learn in this lesson?
• The relationship between a tangent and the radius of a circle .
• The relationship between two tangents segments drawn to a circle from a point outside the circle.
• The relationship between the angle between a tangent and a chord, and the inscribed angle on the opposite side of the chord.
• How to calculate the length of an arc in a circle
• How to calculate the area of a sector in a circleComment calculer l'aire d'un secteur d'un cercle
What is a tangent?
A line that intersects the circle once, and only once, is called a tangent. The point where the line intersects the circle is called point of tangency.
In application: Use the above definition to draw a tangent on the following circle.
tangent
point oftangency
Tangent property (1)
Tangent Theorem, part 1A tangent to a circle is perpendicular to the radius at the point of tangency.
The converse of this theorem is also true.If a line is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle.
In application: Use the above definition to represent the Tangent Theorem part 1 in the following circle.
Tangent property (2)
Tangent Theorem, part 2The tangent segments to a circle from any external points are congruent.
In application: Use the above definition to represent the Tangent Theorem part 2 in the following circle.
Tangent property (3)
Tangent Theorem, part 3The angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord.
In application: Use the above definition to represent the Tangent Theorem part 3 in the following circle.
Example (p. 429, Example 5)1. Find the length of TR. The perimeter is 44 cm, TS = TR and SC = 6 cm.
T
C S
B
RA
Example (p. 430, Example 7)2. Find the measures of and .Given and . A
B
C
D
E
FO
Worksheet
p. 431 #s 114
1. x = 5
2. w = 20o x = 70oy = 9 z = 20o
3. x = 11,2
4. x = 13 y = 13
5. x = 5,3
6. x = 140o
7. 1 = 64o 2 = 71o
8. 1 = 18o 2 = 81o
9. 1 = 30o 2 = 75o
10.1 = 65o 2 = 65o
11.1 = 49o 2 = 61o3 = 70o 4 = 61o 5 = 40o
12.1 = 51o 2 = 39o3 = 39o
13.1 = 77o 2 = 37o3 = 73o 4 = 106o14.1 = 66o 2 = 66o3 = 66o 4 = 57o 5 = 48o
Calculate the missing values.
55o
145o
y
x
z
What are an arc and a sector?
A sector of a circle is a region bounded by two radii and their intercepted arc. The central angle is also known as the sector angle.
An arc of a circle is a section of the circumference of the circle.
In application: Use the above definition to draw a sector, a sector angle, and an arc on the following circle.
sector
sector angle
arc
The length of an arc
Remember the formula for the circumference of a circle:
Since an arc represents a section of the circumference, we multiply the circumference by the proportion of the arc. Therefore, the length of an arc is calculated by using the following formula (where θ is the central angle) :
Example: Calculate the length of the following arc:
The sector area
Remember the formula for the area of a circle:
Since an arc represents a section of the total area, we multiply the total area by the proportion of the sector area. Therefore, the sector area is calculated by using the following formula (where θ is the central angle) :
Example: Calculate the following sector area:
ExampleCalculate the arc length and the sector area for the following circles:
Worksheet
p. 435436 #s 114
Page 435436 #114
1. 12,9 cm
2. 63,4 cm
3. 65,0 cm
4. 261,5 cm
5. 1312,5 cm2
6. 349,1 cm2
7. 7259,5 cm2
8. 4241,2 cm2
9. 87o
10. 108o
11. 36o
12. 320o
13. 224o
14. 16o
Exit slipCalculate the measure of the smallest central angle of a circle where the radius is 10 cm. The circle has a minor arc of 50 cm.
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