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

• Now consider two number lines intersecting Usually but not entirely necessary Denoted by

Cartesian product

21 2 1 2 1 2( , )d x x x x x x

2

Points

• Note coordinate axis from Activity 5.2

• Note non- axes Units on each axis need not be equal

1, 2

Distance• How to determine distance?

• Use Law of Cosines• Then generalize for any two ordered pairs• What happens when = 90 ?

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

,2 2

p q p qx x y y

Lines• A one dimensional object which has both

Location Direction

• Algebraic description must give both

• Matching Slope-intercept General form Intercept form Point-slope

2

0 0

) 1

)

)

)

)

x yA

a bB Ax By C

C y x h k

D y y m x x

E y m x b

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

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 5.3A 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

0 0

1x y

a bAx By C

y y m x x

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 5.4Two lines are parallel iff the two lines have equal slopes

• Proof:Use x-axis as a transversal … corresponding angles

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)

12

1m

m

Perpendicular Lines

Proof

• Use coordinatesand resultsof PythagoreanTheorem forABC

• Also representslopes of AC and CB using coordinates

Distance

• Circle: Locus of points, same distance from fixed

center Can be described by center and radius

2 2r x a y b

Distance

• For given circle with Center at (2, 3) Radius = 5

• Determine equationy = ?

Distance

• Consider the distance between a point and a line What problems

exist?

• Consider thecircle centeredat C, tangentto 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 oftwo lines

Now use distanceformula

Using Coordinates in Proofs

• Consider Activity 5.7

The lengths ofthe three segmentsare equal

• Use equations,coordinates to prove

Using Coordinates in Proofs

• Set one corner at (0,0)

• Establish arbitrarydistances, c and d

• Determine midpointcoordinates

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

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

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

1 2, ,Power P C Power P C

2 2 21 1

2 2 22 2

( ) ( )

( ) ( )

C x a y b r

C x c y d r

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 , 5.5,3

r

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)

2 2 2

tan

x y r

y

x

cos

sin

x r

y r

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 thepoints

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