Analytic Geometry

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)
• 11 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 5.1The 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 5.2For 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?
• 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 5.3A line can be described by a linear
  equation, and a linear equation describes a
  line.
• Author suggestsgeneral 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 5.4Two lines are parallel iff the two
  lines have equal slopes
• ProofUse x-axis as a transversal
  corresponding angles

14
Perpendicular Lines

• Theorem 5.5Two lines (neither vertical) with
  slopes m1 and m2 are perpendicular iff m1 ? m2
  -1
• Equivalent to saying(the slopes arenegative
  reciprocals)

15
Perpendicular Lines

• Proof
• Use coordinatesand resultsof PythagoreanTheorem
  for?ABC
• Also representslopes 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
• Center at (2, 3)
• Radius 5
• Determine equationy ?

18
Distance

• Consider the distance between a point and a line
• What problems exist?
• Consider thecircle centeredat C, tangentto 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 equationof line PQ
• Then determineintersection oftwo lines
• Now use distanceformula

21
Using Coordinates in Proofs

• Consider Activity 5.7
• The lengths ofthe three segmentsare equal
• Use equations,coordinates to prove

22
Using Coordinates in Proofs

• Set one corner at (0,0)
• Establish arbitrarydistances, c and d
• Determine midpointcoordinates

23
Using Coordinates in Proofs

• Determine equationsof the lines AC, DE, FB
• Solve for intersections at G and H
• Use distanceformula to findAH, 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 ofperpendicularsto be the y-axis
• Locate concurrencypoint for the linesat 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
• 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
• Locate these points(3, ?/2), (2, 2?/3),
  (-5, ?/4), (5, -?/3)
• Note
• (x, y) ? (r, ?) is not 11
• (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
• 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 Curvef(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 thepoints

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