1. Lesson 8-1: Circle Terminology

2. Lesson 8-1: Circle Terminology
Circle Definition
Circle :
The set of points coplanar points equidistant from a given point.
The given point is called the CENTER of the circle.
The distance from the center to the circle is called the RADIUS.
Center
Radius
Lesson 8-1: Circle Terminology

3. Lesson 8-1: Circle Terminology
Definitions
Chord :
The segment whose endpoints lie on the circle.
Diameter :
A chord that contains the center of the circle.
Tangent :
A line in the plane of the circle that intersects the circle in exactly one point.
Tangent
Point of Tangency :
Chord
The point where the tangent line intersects the circle.
Diameter
Secant :
A line that contains a chord.
Secant
Lesson 8-1: Circle Terminology

4. Lesson 8-1: Circle Terminology
Example: In the following figure identify the chords, radii, and diameters.
Chords: O D A B F C E
Radii:
Diameter:
Lesson 8-1: Circle Terminology

5. Lesson 8-1: Circle Terminology
Definitions
Congruent Circles :
Circles that have congruent radii. 2 2
Concentric circles :
Circles that lie in the same plane and have the same center.
Lesson 8-1: Circle Terminology

6. Lesson 8-1: Circle Terminology
Polygons
Inscribed Polygon:
A polygon inside the circle whose vertices lie on the circle.
Circumscribed Polygon :
A polygon whose sides are tangent to a circle.
Lesson 8-1: Circle Terminology

7. Lesson 8-1: Circle Terminology
ARCS
Arcs :
The part or portion on the circle from some point B to C is called an arc. Named by 2 letters A B C
Example: B
Semicircle:
An arc that is equal to 180°.
Named by 3 letters O A
Example: C
Lesson 8-1: Circle Terminology

8. Lesson 8-1: Circle Terminology
Minor Arc & Major Arc
Minor Arc :
A minor arc is an arc that is less than 180°
A minor arc is named using its endpoints with an “arc” above. A
Example:
Major Arc:
A major arc is an arc that is greater than 180°. B B O
A major arc is named using its endpoints along with another point on the arc (in order). A
Example: C
Lesson 8-1: Circle Terminology

9. Lesson 8-1: Circle Terminology
Example: ARCS
Identify a minor arc, a major arc, and a semicircle, given that is a diameter.
Minor Arc: A C D E F
Major Arc:
Semicircle:
Lesson 8-1: Circle Terminology

10. Lesson 8-3
Tangents
Lesson 8-3: Tangents

11. THEOREM #1:
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Example:
Find the value of A B C D 4 3 A B C
Lesson 8-3: Tangents

12. THEOREM #2:
If two segments from the same exterior point are tangent to a circle, then they are congruent. B C A
Example:
Find the value of
If AB = 1.8 cm, then AF = 1.8 cm
AE = AF + FE
AE = = 8.8 cm
If FE = 7.0 cm, then DE = 7.0 cm
CE = CD + DE
CE = = 9.4 cm
2.4 cm
1.8
7.0 E A C F B D
Lesson 8-3: Tangents

13. Lesson 8-4: Arcs and Chords

14. Lesson 8-4: Arcs and Chords
Theorem #1:
In a circle, if two chords are congruent then their corresponding minor arcs are congruent. E A B C D
Example:
Lesson 8-4: Arcs and Chords

15. Lesson 8-4: Arcs and Chords
Theorem #2:
In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc. E D A C B
Example:
If AB = 5 cm, find AE.
Lesson 8-4: Arcs and Chords

16. Lesson 8-4: Arcs and Chords
Theorem #3:
In a circle, two chords are congruent if and only if they are equidistant from the center. O A B C D F E
Example:
If AB = 5 cm, find CD.
Since AB = CD, CD = 5 cm.
Lesson 8-4: Arcs and Chords

17. Lesson 8-4: Arcs and Chords
Try Some Sketches:
Draw a circle with a chord that is 15 inches long and 8 inches from the center of the circle.
Draw a radius so that it forms a right triangle.
How could you find the length of the radius?
Solution:
∆ODB is a right triangle and
8cm
15cm O A B D x
Lesson 8-4: Arcs and Chords

18. Lesson 8-4: Arcs and Chords
Try Some Sketches:
Draw a circle with a diameter that is 20 cm long.
Draw another chord (parallel to the diameter) that is 14cm long.
Find the distance from the smaller chord to the center of the circle.
Solution:
10 cm
20cm O A B D C
∆EOB is a right triangle.
OB (radius) = 10 cm
14 cm E x
7.1 cm
Lesson 8-4: Arcs and Chords

19. Lesson 8-5: Angle Formulas
Angle in Circles
Lesson 8-5: Angle Formulas

20. Lesson 8-5: Angle Formulas
Central Angle
Definition:
An angle whose vertex lies on the center of the circle.
Central Angle
(of a circle)
Central Angle
(of a circle)
NOT A Central Angle
(of a circle)
Lesson 8-5: Angle Formulas

21. Lesson 8-5: Angle Formulas
Central Angle Theorem
The measure of a center angle is equal to the measure of the intercepted arc. Y Z O
110
Intercepted Arc
Center Angle
Example:
Give is the diameter, find the value of x and y and z in the figure.
Lesson 8-5: Angle Formulas

22. Example: Find the measure of each arc.
4x + 3x + (3x +10) + 2x + (2x-14) = 360°
14x – 4 = 360°
14x = 364°
x = 26°
4x = 4(26) = 104°
3x = 3(26) = 78°
3x +10 = 3(26) +10= 88°
2x = 2(26) = 52°
2x – 14 = 2(26) – 14 = 38°
Lesson 8-5: Angle Formulas

23. Lesson 8-5: Angle Formulas
Inscribed Angle
Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle).
Examples: 3 1 2 4
No!
Yes!
No!
Yes!
Lesson 8-5: Angle Formulas

24. Lesson 8-5: Angle Formulas
Intercepted Arc
Intercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds:
1. The endpoints of the arc lie on the angle.
2. All points of the arc, except the endpoints, are in the interior of the angle.
3. Each side of the angle contains an endpoint of the arc.
Lesson 8-5: Angle Formulas

25. Lesson 8-5: Angle Formulas
Inscribed Angle Theorem
The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Y
Inscribed Angle
110
55 Z
Intercepted Arc
An angle formed by a chord and a tangent can be considered an inscribed angle.
Lesson 8-5: Angle Formulas

26. Lesson 8-5: Angle Formulas
Examples: Find the value of x and y in the fig. y x 50 A B C E F y 40 x 50 A B C D E
Lesson 8-5: Angle Formulas

27. Lesson 8-5: Angle Formulas
An angle inscribed in a semicircle is a right angle. P
180
90 S R
Lesson 8-5: Angle Formulas

28. Interior Angle Theorem
Definition:
Angles that are formed by two intersecting chords. 1 A B C D 2 E
Interior Angle Theorem:
The measure of the angle formed by the two intersecting chords is equal to ½ the sum of the measures of the intercepted arcs.
Lesson 8-5: Angle Formulas

29. Lesson 8-5: Angle Formulas
Example: Interior Angle Theorem
91 A C x° y° B D
85
Lesson 8-5: Angle Formulas

30. Lesson 8-5: Angle Formulas
Exterior Angles
An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. 3 y x 2 1
Two secants
2 tangents
A secant and a tangent
Lesson 8-5: Angle Formulas

31. Exterior Angle Theorem
The measure of the angle formed is equal to ½ the difference of the intercepted arcs.
Lesson 8-5: Angle Formulas

32. Example: Exterior Angle Theorem
Lesson 8-5: Angle Formulas

33. Lesson 8-5: Angle FormulasQ G F D E C 1 2 3 4 5 6 A
30°
25°
100°
Lesson 8-5: Angle Formulas

34. Inscribed Quadrilaterals
If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.
mDAB + mDCB = 180 
mADC + mABC = 180 
Lesson 8-5: Angle Formulas

35. Lesson 8-6: Segment Formulas
Segments in Circles
Lesson 8-6: Segment Formulas

36. Intersecting Chords Theorem
Interior segments are formed by two intersecting chords.
Theorem:
If two chords intersect within a circle, then the product of the lengths of the parts of one chord is equal to the product of the lengths of the parts of the second chord. A B C D E a d b c
a • b = c • d
Lesson 8-6: Segment Formulas

37. Intersecting Secants/Tangents
Exterior segments are formed by two secants, or a secant and a tangent. D B C A D B A C E
Secant and a Tangent
Two Secants
Lesson 8-6: Segment Formulas

38. Intersecting Secants Theorem
If two secant segments are drawn to a circle from an external point, then the products of the lengths of the secant and their exterior parts are equal.
a • e = c • f
Lesson 8-6: Segment Formulas

39. Lesson 8-6: Segment Formulas
Example:
AB  AC = AD  AE D B A C E
4 cm
4  10 = 2  (2+x)
6 cm
2 cm
40 = 4 + 2x x
36 = 2x
X = 18 cm
Lesson 8-6: Segment Formulas

40. Secant and Tangent Theorem:
The square of the length of the tangent equals the product of the length of the secant and its exterior segment. D B C A
a2 = b • d a b c d
Lesson 8-6: Segment Formulas

41. Lesson 8-6: Segment Formulas
Example: D B C A x
9 cm
25 cm
Lesson 8-6: Segment Formulas