Analytic Geometry

Analytic GeometryRene Descartes1596 1650 Considered as a father of Analytic GeometryFrench mathematician introduced the concept of coordinate system Le Geometrie unified the branches of Algebra and Geometry Distance Directed line line with a direction extending in both the positive and the negative directionDirected line segment the segment between the line consisting two points. Directed distance Directional Undirected distance Absolute Value of the directed distance ABCartesian PlaneRectangular Coordinate system Intersection of two number lines Origin (0, 0) X axis and Y axis Ordered pairs( x , y )AbcissaOrdinateDistance between two pointsDivision of a line segment General formula given r is the ratio: Dividing by the midpoint: Division of a line segment A point P(x, y) is on the line through A( 4, 4) and B(5, 2). Find the coordinates of P given that the segment AB is extended through B to P so that P is twice as far from A as from B Given the points on the number 2, find the coordinates of P given that segment AB is extended through A to P so that P is three times as far from B as from A

Division of a line segment 3. Find the coordinates of the midpoint of the sides of each triangle whose vertices are (1, 2) (2, 5) and (6, 3)4. Find the coordinates of the point which divides the line segment from ( 1, 4) to (2, 3) in the ratio 3 to 4. 5. Find the point of the line passing through A( 1, 1) and B(4, 4) which is twice as far from A as from B (two cases)

Division of a line segment Note: The point P is between A and B if and only if 0 < r < 1. However, if P is a point on the segment AB extended through B, then the length of the segment AP is greater than the length of segment AB, thus r becomes greater than 1.

InclinationThe inclination of a line that intersects the x-axis is the smallest angle greater than or equal to 0, that the line makes with the positive direction of the x-axis. The inclination of a horizontal line is o.

Slope of the lineThe slope of the line is the tangent of the inclination. The rise over the run

Inclined to the right positive slopeInclined to the left negative slope

Inclination and slopeDraw the line through P (2, 2) with an inclination of 35Draw the line through the point P ( 2, 2) with the slope of 2/3

Slope of the lineSlope = m = tan

ExamplesGiven the points A( 1, 1), B( 5, 0), C(4, 3) and D ( 2, 2). Show that ABCD is a parallelogram Verify if each triangle with the given points A(4, 4), B(4, 4) and C(0,0) are the vertices of a right triangle

Slope of the lineA cross section of a cottage, 18 feet wide is an isosceles triangle. If the slope of a side is 1.75 and there is a second floor 8 feet above the ground floor, what is the width of the second floor?

Given points P (0, 1), Q (9, 6) and R ( 4, 1), verify if the three points lie on a straight line.

Slope of the line3. Two lines passing through (3, 2) make an angle of 45. If the slope of one of the lines is 1, find the slope of the other line.

4. A flat board leans against a wall. The upper edge is 6ft above the ground and the lower edge is 2ft out from the wall. What is the slope of the board?

TheoremTwo non-vertical lines are parallel if and only if their slopes are equalThe lines that are perpendicular will have their slopes as their negative reciprocals.

Angle between two linesm2 is the slope of the terminal sidem1 is the slope of the terminal sideCounterclockwise movementA television camera is located along the 40-yd line at a football game. If the camera is 20 yds back from the sideline, through what angle should it be able to pan in order to cover the entire field of play, including end zones, which are 10 yds deep?Angle between two lines

Angle between two linesArea of any polygon (x1, y1) and (x2, y2) is moving in a counterclockwise directionArea of any polygon Find the area of the triangle with vertices of (4, 3); ( 2, 5), (2, 3)

Find the Area of Triangle CBA with vertices C (3, 0); B ( 2, 1) and A (1, 2)Area of any polygon 3. Find the area of the pentagon whose vertices are (2, 3); ( 1, 2); (3, 0); (0, 8) and (2, 1)

4. Find the area of the quadrilateral whose vertices are (2, 3), (6, 5), (4, 2) and (4, 0)