1. Analytic Geometry
Chapter 5

2. Analytic Geometry Unites geometry and algebra
Coordinate system enables
Use of algebra to answer geometry questions
Proof using algebraic formulas
Requires understanding of points on the plane

3. Points Consider Activity 5.1 Number line
Positive to right, negative left (by convention)
1:1 correspondence between reals and points on line
Some numbers constructible, some not (what?)

4. Points Distance between points on number line
Now consider two number lines intersecting
Usually  but not entirely necessary
Denoted by Cartesian product

5. Points Note coordinate axis from Activity 5.2 Note non-  axes
Units on each axis need not be equal

6. Distance How to determine distance? Use Law of Cosines
Then generalize for any two ordered pairs
What happens when  = 90 ?

7. Midpoints
Theorem 5.1 The midpoint of the segment between two points P(xP, yP) and Q(xQ, yQ) is the point
Prove
For non  axes
For  axes

8. Lines A one dimensional object which has both
Location
Direction
Algebraic description must give both
Matching
Slope-intercept
General form
Intercept form
Point-slope

9. Slope
Theorem 5.2 For a non vertical line the slope is well defined. No matter which two points are used for the calculation, the value is the same

10. Slope What about a vertical line? Should not say slope is infinite
The x value is zero
The slope is undefined
Should not say slope is infinite
Positive? Negative?
Actually infinity is not a number

11. Linear Equation
Theorem 5.3 A line can be described by a linear equation, and a linear equation describes a line.
Author suggests general form is most versatile
Consider the vertical line

12. Alternative Direction Description
Consider Activity 5.4
Specify direction with angle of inclination
Note relationship between slope and tan 
Consider what happens with vertical line

13. Parallel Lines
Theorem 5.4 Two lines are parallel iff the two lines have equal slopes
Proof: Use x-axis as a transversal … corresponding angles

14. Perpendicular Lines
Theorem 5.5 Two lines (neither vertical) with slopes m1 and m2 are perpendicular iff m1  m2 = -1
Equivalent to saying (the slopes are negative reciprocals)

15. Perpendicular Lines Proof
Use coordinates and results of Pythagorean Theorem for ABC
Also represent slopes of AC and CB using coordinates

16. Distance Circle: Locus of points, same distance from fixed center
Can be described by center and radius

17. Distance For given circle with Determine equation y = ?
Center at (2, 3)
Radius = 5
Determine equation y = ?

18. Distance Consider the distance between a point and a line
What problems exist?
Consider the circle centered at C, tangent to the line

19. Distance
Constructing the circle
Centered at C
Tangent to the line

20. Using Analysis to Find Distance
Given algebraic descriptions of line and point
Determine equation of line PQ
Then determine intersection of two lines
Now use distance formula

21. Using Coordinates in Proofs
Consider Activity 5.7
The lengths of the three segments are equal
Use equations, coordinates to prove

22. Using Coordinates in Proofs
Set one corner at (0,0)
Establish arbitrary distances, c and d
Determine midpoint coordinates

23. Using Coordinates in Proofs
Determine equations of the lines AC, DE, FB
Solve for intersections at G and H
Use distance formula to find AH, HG, and GC

24. Using Coordinates in Proofs
Note figure for algebraic proof that perpendiculars from vertices to opposite sides are concurrent (orthocenter)
Arrange one of perpendiculars to be the y-axis
Locate concurrency point for the lines at x = 0

25. Using Coordinates in Proofs
Recall the radical axis of two circles is a line
We seek points where
We calculate
Set these equal to each other, solve for y

26. Polar Coordinates Uses Describe a point P by giving Point P is
Origin point
Single axis (a ray)
Describe a point P by giving
Distance to the origin (length of segment OP)
Angle OP makes with polar axis
Point P is

27. Polar Coordinates Try it out Note
Locate these points (3, /2), (2, 2/3), (-5, /4), (5, -/3)
Note
(x, y)  (r, ) is not 1:1
(r, ) gives exactly one (x, y)
(x, y) can be many (r, ) values

28. Polar Coordinates Conversion formulas From Cartesian to polar
Try (3, -2)
From polar to Cartesian
Try (2, /3)

29. Polar Coordinates Now Use these to convert Try 3x + 5y = 2
Ax + By = C to r = f()
Try 3x + 5y = 2
Convert to polar equation
Also r sec  = 3
Convert to Cartesian equation

30. Polar Coordinates Recall Activity 5.11 Shown on the calculator
Graphing y = sin (6)

31. Polar Coordinates Recall Activity 5.11 Change coefficient of 
Graphing y = sin (3)

32. Polar In Geogebra Consider graphing r = 1 + cos (3)
Define f(x) = 1 + cos(3x)
Hide the curve that appears.
Define Curve[f(t) *cos(t), f(t) *sin(t), t, 0, 2 * pi]

33. Polar In Geogebra Consider these lines They will display polar axes
Could be made into a custom tool

34. Nine Point Circle, Reprise
Recall special circle which intersects special points
Identify the points

35. Nine Point Circle
Circle contains …
The foot of each altitude

36. Nine Point Circle
Circle contains …
The midpoint of each side

37. Nine Point Circle Circle contains …
The midpoints of segments from orthocenter to vertex

38. Nine Point Circle Recall we proved it without coordinates
Also possible to prove by
Represent lines as linear equations
Involve coordinates and algebra
This is an analytic proof

39. Nine Point Circle Steps required Place triangle on coordinate system
Find equations for altitudes
Find coordinates of feet of altitudes, orthocenter
Find center, radius of circum circle of pedal triangle

40. Nine Point Circle Steps required
Write equation for circumcircle of pedal triangle
Verify the feet lie on this circle
Verify midpoints of sides on circle
Verify midpoints of segments orthocenter to vertex lie on circle

41. Analytic Geometry
Chapter 5