1.0_cartesianplane

UNIT IThe Cartesian Plane

EM 121: Analytic and Solid Geometry

References:• Chapter1&2: Analytic Geometry by G. Fuller

• Appendix D: Calculus with Analytic Geometry by R. Larson• Chapter2: College Algebra by R. D. Gustafson


Unit I: The Cartesian Plane Rectangular or Cartesian Coordinates of a Poi

nt Distance Formula Midpoint Formula Gradient or Slope of a Line Equations of a Straight Line Directed Distance from a Line to a Point Parallel and Perpendicular Lines Angle between Two Lines Applications

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Rectangular or Cartesian Coordinates of a Point

Unit I: The Cartesian Plane

EM 121: Analytic and Solid Geometry 4


ordered pair (x, y)

Rectangular coordinate system / Cartesian plane model for representing ordered pairs French mathematician, René Descartes▪ It is developed by considering two real lines intersecting

at right angles Co-invent with another Frenchman, Pierre Fermat

EM 121: Analytic and Solid Geometry 5


x-axis horizontal real line

y-axis vertical real line

origin point of intersection

two axes divide the plane into four quadrants



coordinates Each point in the plane is identified by an ordered

pair (x, y)

the point (x, y) the first coordinate is the x-coordinate or abscissa,

and the second coordinate is the y-coordinate or ordinate

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Distance FormulaUnit I: The Cartesian Plane


The Distance Formula

Recall: Pythagorean Theorem right triangle hypotenuse c and sides a and b are related by

a2 + b2 = c2

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The Distance Formula

determine the distance d between the two points P(x1 , y1) and R(x2 , y2)

By the Pythagorean Theorem,

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Example

1) Find the distance between the points P(-2, 1) and Q(3, 4)

2) Verify that the points P(2, 1), Q(4, 0) and R(5, 7) form the vertices of a right triangle.

3) Find x such that the distance between P(x,3) and R(2, -1) is 5.

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Midpoint FormulaUnit I: The Cartesian Plane


Midpoint Formula

The coordinates of the midpoint of the line segment joining two points can be found by “averaging” the x-coordinates of the two points and “averaging” the y-coordinates of the two points.

the midpoint of the line segment joining the points (x1 , y1) and (x2 , y2) in the plane is

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Example

1. Find and draw the Midpoint M of the line segment (-5, -3) and (9, 3)

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Gradient or Slope of a Line



Gradient of a Line

The degree of steepness of the slope of the line if it is given by vertical & horizontal change

If the angle between the line and x-axis , is known then,

16

tanm

12

12

xx

yy

x

ym


Gradient of a Line

Various Type of Gradient

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Gradient of a Line

Various Type of Gradient

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Example

1. Given the points A(-1, 3) and B(2, 4) a) Distance between A and Bb) Midpoint of A and Bc) Gradient of Line Segment A and B

2. Given the vertices of a are A(-1, 0), B (5,2) and C (3,-2). Find

a) The lengths of the sides of ABCb) The midpoint of the line segment joining A and Cc) The slope of a line segment BC and ACd) The three points which divide the line segment joining A

and B into four parts19


Practice Problems

1. The midpoint of the line segment between P1 (x, y) and P2 (-2, 4) is PM (2, -1). Find the coordinates of P1.

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Equations of Straight Lines



Equation of a Straight Line Definition:

A straight line is a set of all points satisfying the general equation.

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0 CByAx where A, B , C are real numbers


Equation of a Straight Line

Several Forms of Equation of a Line1)

2)

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if the line parallel ( // ) to y-axis and passes through point (x, y)

1xx

1yy parallel to x-axis and passes through point (x , y)

y

x

L

(x1 , 0)

x = x1

x1O

y

x

L(0, y1)

y = y1y1

O


Equation of a Straight Line3) Point-Slope Form

4) Two Point Form

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11 xxmyy if the point of the line and its slope(m) are known

1

12

121 xx

xx

yyyy

If the two points on the line are known


Equation of a Straight Line5) Slope and Intercept

6) Intercept Form

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1b

y

a

x if x and y intercept are known

x

y

L

(0, b)

(a, 0)O

1b

y

a

x

bmxy if the slope (m) and the y-intercept are known


Example

1. Find the Equation of the line with the given requirements:a) Slope 4 and thru (1, -4)b) Passing thru (5, 2) and (-1, -6)c) Slope 0 and passing thru (-2, -7)d) Slope -3/2 and passing thru the midpoint of the

line segment joining (2, -3) and (5, -1)e) Slope -4/5 and y-intercept 3f) // to y-axis and passing thru (4, -3)g) x-intercept is 4 and y-intercept is -1

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Directed Distance from a Line to a Point




Definition: The directed distance from line Ax + By + C = 0 to

the point P1(x1, y1) is given by the formula

where the denominator is given the sign of B. The distance is positive if the point P1 is above the line, and negative if P1 is below the line.

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22

11

BA

CByAxd



1. Find the distance from the line 5x = 12y + 26 to the points P1(3, -5), P2(-4, 1), and P3(9, 0).

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Seatwork(by pair)1whole sheet of paper

1. How far is the line 3x-4y +5 = 0 from the origin.

2. Find the slope and intercepts of line represented by the equation 3x – 4y = 12

3. Find the Equation of the line with the given requirements:a) Slope -4 and y-intercept 5b) Passes thru point A(-3, -6) with slope -1/2c) Passing thru points A(3, -2) and B(3, 7)

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Parallel and Perpendicular Lines



Parallel and Perpendicular Lines Parallel Lines

Definition:▪ Two lines are said to be parallel (//) if their slopes are

equal

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21 mm


Parallel and Perpendicular Lines Perpendicular Lines

Definition:▪ Two lines with slopes m1 and m2 are perpendicular () if

and only if ; that is, if their slopes are negative reciprocals

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12

1

mm

121 mm


Example

1. Check to see whether the line through pts P1 and Q1 is // or to the line through P2 and Q2 from the given points.a) P1(3, 1), Q1 (-2, 7) and P2(5, -3), Q2 (-1, -8)

b) P1(12, 8), Q1 (4, 8) and P2(3, 1), Q2 (-6, 1)

2. Find the equation of the line in general form that passes through the (4, 1), and is perpendicular to the line 2x – 3y + 4 = 0.

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Angle between Two Lines



Angle between Two Lines

Theorem: If φ is an angle, measured

counterclockwise, between two lines, then

where m2 is the slope of the terminal side and m1 is the slope of the initial side

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21

12

1tan

mm

mm

L1

L2

y

xO

Ψ

φ

θ1 θ2α

Sum of all interior angles is equal to 180180 = θ1 + φ + α α = 180- θ2

180= θ1 + φ + (180- θ2)Φ = θ2 - θ1


Tangent of the difference of two angles:

tan Φ = tan(Φ2 –Φ1) = (tanΦ2 – tanΦ1)/ (1+tanΦ1tanΦ2)

tanΦ = (m2 – m1)/ (1+m1m2)

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Example

Find the tangents of the angles of the triangle whose vertices are A(3, -2), B(-5, 8) & C(4, 5). Express each angle to the nearest degree.

Find the acute angle between the two lines that have m1, = 3 and m2 = 7 for their slopes.

Find the obtuse angle between the X axis and a line with a slope of m = - 8.

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ApplicationsUnit I: The Cartesian Plane

EM 121: Analytic and Solid Geometry39


Applications

Predicting stock prices. The value of the stock of ABC Corporation has been increasing by the same fixed dollar amount each year. The pattern is expected to continue. Let 2008 be the base year corresponding to x = 0 with x = 1, 2, 3, … corresponding to later years. ABC stock was selling at $37½ in 2008 and at $45 in 2010. If y represents the price of ABC stock, find the equation y = mx + b that relates x and y, and predict the price in the year 2012.

Fun Run. From the start, a man runs 4km towards north, 3km towards east, and takes a rest. He runs 6km towards south and finishes 2km towards east. Draw the line from starting point to resting point and a line from resting point to finish point. Find the angle between the lines assuming that east-west is parallel to x-axis.

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