Analytic Geometry
Chapter 5
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
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?)
Points
• Distance between points on number line
d ( x1 , x2 )  x1  x2 
 x1  x2 
2
• Now consider two number lines
intersecting
 Usually  but not entirely necessary
 Denoted by
   
2
Cartesian product
Points
• Note coordinate axis from Activity 5.2
1, 2 
• Note non-  axes
 Units on each axis need not be equal
Distance
• How to determine distance?
• Use Law of Cosines
• Then generalize for any two ordered pairs
• What happens when  = 90 ?
Midpoints
• Theorem 5.1
The midpoint of the segment between two
points P(xP, yP) and Q(xQ, yQ) is the point
 x p  xq y p  yq 
,


2 
 2
• Prove
 For non  axes
 For  axes
Lines
• A one dimensional object which has both
 Location
 Direction
• Algebraic description must give both
x y
• Matching
A)
 1
B)
a b
Ax  By  C
 Intercept form
C)
y   x  h  k
 Point-slope
D)
y  y0  m   x  x0 
E)
y  m x b
 Slope-intercept
 General form
2
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
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
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
x y
 1
a b
Ax  By  C
y  y0  m   x  x0 
y  m x b
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
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
Perpendicular Lines
• Theorem 5.5
Two lines (neither vertical) with slopes m1
and m2 are perpendicular iff m1  m2 = -1
 Equivalent to saying
1
m1  
(the slopes are
m2
negative reciprocals)
Perpendicular Lines
Proof
• Use coordinates
and results
of Pythagorean
Theorem for
ABC
• Also represent
slopes of AC and CB using coordinates
Distance
• Circle:
 Locus of points, same distance from fixed
center
 Can be described by center and radius
r
 x  a   y  b
2
2
Distance
• For given circle with
 Center at (2, 3)
 Radius = 5
• Determine equation
y=?
Distance
• Consider the distance between a point and
a line
 What problems
exist?
• Consider the
circle centered
at C, tangent
to the line
Distance
• Constructing the circle
 Centered at C
 Tangent to
the line
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
Using Coordinates in Proofs
• Consider Activity 5.7
 The lengths of
the three segments
are equal
• Use equations,
coordinates to prove
Using Coordinates in Proofs
• Set one corner at (0,0)
• Establish arbitrary
distances, c and d
• Determine midpoint
coordinates
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
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
Using Coordinates in Proofs
• Recall the radical axis of two circles is a
line
• We seek points where
Power  P, C1   Power  P, C2 
• We calculate

C1  ( x  a)2  ( y  b)2  r12
C2  ( x  c)  ( y  d )  r
2
2
2
2
• Set these equal to each other, solve for y
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


 r ,    5.5, 
3

Polar Coordinates
• Try it out
 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
Polar Coordinates
• Conversion formulas
 From Cartesian to polar
 Try (3, -2)
 From polar to Cartesian
 Try (2, /3)
 x2  y 2  r 2


y
 tan  
x

 x  r cos 

 y  r sin 
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
Polar Coordinates
• Recall Activity 5.11
 Shown on the calculator
 Graphing y = sin (6)
Polar Coordinates
• Recall Activity 5.11
 Change coefficient of 
 Graphing y = sin (3)
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]
Polar In Geogebra
• Consider these lines
• They will display
polar axes
 Could be made
into a custom tool
Nine Point Circle, Reprise
• Recall special circle which intersects
special points
• Identify the
points
Nine Point Circle
• Circle contains …
 The foot of each altitude
Nine Point Circle
• Circle contains …
 The midpoint of each side
Nine Point Circle
• Circle contains …
 The midpoints of segments from orthocenter
to vertex
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
Nine Point Circle
Steps required
1.Place triangle on coordinate system
2.Find equations for altitudes
3.Find coordinates of feet of altitudes,
orthocenter
4.Find center, radius of circum circle of
pedal triangle
Nine Point Circle
Steps required
5.Write equation for circumcircle of pedal
triangle
6.Verify the feet lie on this circle
7.Verify midpoints of sides on circle
8.Verify midpoints of segments orthocenter
to vertex lie on circle
Analytic Geometry
Chapter 5