Analytic Geometry Chapter 4
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 4.1 • 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 4.2 • 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 4.1The 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 4.2For 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 4.2A 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
Alternative Direction Description • Consider Activity 4.4 • Specify direction with angle of inclination • Note relationship between slope and tan • Consider what happens with vertical line
Parallel Lines • Theorem 4.4Two lines are parallel iff the two lines have equal slopes • Proof:Use x-axis as a transversal … corresponding angles
Perpendicular Lines • Theorem 4.5Two lines (neither vertical) with slopes m1and m2are perpendicular iff m1 m2 = -1 • Equivalent to saying(the slopes arenegative reciprocals)
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
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 equationof line PQ • Then determineintersection oftwo lines • Now use distanceformula
Using Coordinates in Proofs • Consider Activity 4.7 • The areas of thetwo pairs oftriangles areequal • Use equations,coordinates to prove
Using Coordinates in Proofs • We focus on APB and APD • Easy to calculate are of APB • For APD • Let AD be the base, what is the equation? • Now use distanceformula (point to line)to get area of APD • Now simplify and work for equality
Using Coordinates in Proofs • Note we used origin (0, 0) as one of important points • Sometimesis an advantageto use pointsother than origin
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
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
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)
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 4.11 • Shown on the calculator • Graphing y = sin (6)
Polar Coordinates • Recall Activity 4.11 • 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 • 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 4
