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

Points Note coordinate axis from Activity 5.2 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 Prove For non  axes For  axes

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

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? 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

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

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 (the slopes are 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

Distance For given circle with Determine equation y = ? 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 We calculate Set these equal to each other, solve for y

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

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

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

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

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 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

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

Analytic Geometry Chapter 5