Precal 1 Basic Concepts and Straight Lines PDF

PRECAL 1  2  Analytic geometry- deals with the solution of geometric problems by the use of algebraic methods and the interpretation of algebraic equations geometrically as lines, curves or surfaces.  Rene Descartes “Father of Analytic Geometry” ...

PRECAL 1  2  Analytic geometry- deals with the solution of geometric problems by the use of algebraic methods and the interpretation of algebraic equations geometrically as lines, curves or surfaces.  Rene Descartes “Father of Analytic Geometry”  3  One of the basic concepts of analytic geometry is the representation of all real numbers by points on a directed line.  Thereal numbers consist of the positive numbers, the negative numbers, and zero.  4  Rectangular Coordinates  We draw a horizontal line and a vertical line meeting at the origin O.  5 A pair of numbers (x, y) in which the order of occurrence of the numbers is distinguished is an ordered pair of numbers.  The x-coordinate, or abscissa, of a point P is the directed distance from the y-axis to the point.  The y-coordinate, or ordinate, of a point P is the directed distance from the x-axis to the point.  6  Distance Between Two Points  The length of a horizontal line segment joining two points is the abscissa of the point on the right minus the abscissa of the point on the left.  7  Distance Between Two Points  The length of a vertical line segment joining two points is the ordinate of the upper point minus the ordinate of the lower point.  8  Distance Between Two Points  The length of a slant line segment joining two points is the square root of the sum of the squares of the difference of the abscissas and the difference of the ordinates.  9 DISTANCE FORMULA: d x 2  x1 2  y 2  y1 2  10  Example 1 (horizontal line) Find the distance between points P1 ( 4, 0) and P2 ( 16, 0).  Example 2 (vertical line) Find the distance AB and BC if the points A, B and C line on the same vertical line.  11  Example 3 (slant line) Find the distance between points P1 ( 4, 1) and P2 ( 7, -3)  The Distance Formula Example: Find the distance between points P1 ( 0, -8) and P2 ( 16, 7). Example: Draw the triangle having the vertices A(6, 2), B(2, -3), and C(-2, 2). Show that the triangle is isosceles. The Distance Formula  The Distance Formula Three given points are collinear if; the sum of the two shortest distances is equal to the longest distance.  15 1. Draw the triangle with the given vertices and show that the triangle is a right triangle.  A(1, 3), B(10, 5), and C(2, 1)  16 2. Determine whether the points A(3, 3), B(0, 1), and C(9, 7) lie on a straight line. 3. Find the diameter of a circle with center at (2,3) and passing through the point ( -3, 1)  17 1. One end of a line segment is the point (0,4) and the abscissa of the other end is 6. If the length of the line is 10, find the ordinate of the other end. 2. Find the perimeter and area of a triangle whose vertices are (-1,-2), (-3,0) and (5,4). 3. Find the distance between two points: a. ( 0,-3) and ( 5, -15) b. (1/2, ¾) and ( -8/5, 4/7)  18 4. Draw the triangle with the given vertices and show that the triangle is a right triangle.  A(-1, 1), B(6, -2), and C(4, 3) 6. Determine if the points P(3,2), Q(4,6) and R(0,-8) are collinear or not.  19  slope A builder refers to this as pitch of the roof  20  slope Inclination of an airplane  21  The slope m of a line passing through two given points P1 and P2 is equal to the difference of the ordinates divided by the difference of the abscissas taken in the same order; that is y 2  y1 m x 2  x1 22  23 Zero Slope Undefined Slope y y m is undefined m=0 x x Line is horizontal. Line is vertical.  24  Angle of inclination:  The 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 0.  The slope of a line is the tangent of the inclination.  25  Inclination of Line, θ  The inclination of the line, θ, is the smallest positive angle measured from the positive axis to the line.  26  Inclination of Line, θ Positive Slope Negative Slope y y m>0 m