1. Circles

2. Vocabulary Geometry 12.1 Interior Exterior Radius Diameter Chord
Tangent
Secant

3. Circles Congruent versus Similar Concentric Tangent Internally tangent
Externally tangent

4. Tangents Tangent Perpendicular to radius
Example: Visible distance to horizon from Mt Everest
Mt Everest is 29,000 ft above sea level
Find: EH
2 tangents from external point
Same measure

5. Arcs & Chords Geometry 12.2 Arc Finding Measure: Minor Arc Major Arc
Semi-Circle
Finding Measure:

6. Arcs Adjacent Arcs Congruent Arcs – 2 Arcs with the same measure
Central Angles
Chords

7. Congruent Arcs
Examples
Find RT
Find mCD

8. Radii & Chords If radius is perpendicular to chord
Bisects Chord & Arc
A perpendicular bisector of chord is a radius

9. Radii & Chords
Example: Find QR

10. Examples

11. Inscribed Angles Geometry 12.4
Vertex on circle
Sides contain chords
Measure of inscribed angle = ½ measure of arc
m<E = ½(mDF)

12. Inscribed Angles If inscribed angle arcs are congruent
Intercept same arc or congruent arcs
THEN: Inscribed angles are congruent

13. Example
Find mBC
Find m<ECD
Find m<DEC

14. Example 2: Hobby Application
An art student turns in an abstract design for his art project.
Find mDFA.
mDFA = mDCF + mCDF
Ext  Thm.
Inscribed  Thm.
Substitute.
Simplify.
= 115°

15. Inscribed Angle
Inscribed angle subtends a semicircle if and only if the angle is a right angle
Example:

16. Quadrilaterals
Opposite Angles Add to 180

17. Example: Finding Angle Measures in Inscribed Quadrilaterals
Find the angle measures of GHJK.
Step 1 Find the value of b.
mG + mJ = 180
GHJK is inscribed in a .
3b b + 20 = 180
Substitute the given values.
9b + 45 = 180
Simplify.
9b = 135
Subtract 45 from both sides.
b = 15
Divide both sides by 9.

18. Angle Relationships Geometry 12.5
Tangent and a secant/chord
Measure of angle is ½ intercepted arc measure
Measure of the arc is twice the measure of angle
Example:
Find m<BCD
Find measure of arc AB

19. Interior Angle Intersect inside circle Example:
Measure of vertical angles is ½ sum of arcs
Example:
Find m<PQT

20. Exterior Angle

21. Examples
Find x

22. Lesson Quiz:
4. Find mCE.
12°

23. Segment Relationships Geometry 12.6J
Example: J

24. External Secant Relationships
Example:

25. Sectors & Arc Length Geometry 12.3
Sector of a circle – 2 radii & arc
The pie shaped slice of the circle
Area of sector is percent area of circle based on arc or central angle:
A= Area, r= radius, m= measure of arc/angle

26. Segments of Circles Segment of circle Finding Area of Segment
Area of arc bounded by chord
Finding Area of Segment

27. Arc Length Distance along the arc (circumference) Example:
Measured in linear units
L= length, r= radius, m= measure of arc/angle
Example:
Measure of GH
file://localhost/Users/cmidthun/Downloads/practice_a (39).doc

28. Examples
Sector:
Segment:
segment RST

29. Equation for Circle Geometry 12.7
(x – h)2 + (y – k)2 = r2
h is the x coordinate of the center point
k is the y coordinate of the center point
r is the radius
To determine if point is inside/on/outside
Plug x and y of point into circle equation
h & k are the CENTER POINT coordinates
Compare result to r2
If < pt is inside : if > pt is outside : if = pt is on

30. Finding center & radius
Given 2 endpoints
Find center point
X coordinate is (x1+x2)/2
Y coordinate is (y1+y2)/2
Use center point coordinate and one end point with the distance formula to find the radius
Plug center point and radius into equation for circle

31. Slope of Tangent Slope of radius = Rise over Run (ΔY ÷ ΔX)
Find negative reciprocal
Change sign, flip fraction
Insert negative reciprocal into slope formula
Y = mx + b
Substitute y & x coords from tangent point to find b
Rewrite equation with y & x and the b value

32. Examples with graph paper