Basic Pre-Calculus 1 only for engineering studensts

for reference only

1. Pre calculus Analytic Geometry - deals with geometry problem using coordinates system and converting it into algebraic problems René Descartes RENE DESCARTES (1596 – 1650 , Cartesius in latin ) is regarded as the founder of analytic geometry by introducing coordinates system in 1637. Rectangular Coordinate System – the rectangular coordinate system is a two-dimensional plane consisting of a horizontal axis (x-axis) and a vertical (y-axis). This is also known as Cartesian Coordinate System . Point 0 is the origin and has coordinates (0,0). The x-coordinates or abscissa is always measured from the y-axis while the y-coordinates or ordinates is always measured from the x-axis.

2. Distance Between Two Points in a Plan – consider two points whose coordinates are (X1, Y1) and (X2,Y2) respectively. A right triangle is formed with the distance between two points being the hypotenuse of the right triangle. Pythagorean theorem, the distance between two points can be calculated using : d^2 = (x2-x1)^2 + (y2-y1)^2 Lines – is defined as the shortest distance between two points SLOPE OF A LINE – is defined as the rise (vertical) per run (horizontal) Example 1: Find the distance between point A(4,-3) and B(-2,5). Ans. d=10 Example 2: A point P (X,2) is equidistant from the point (-2,9) and (4,-7). What is the value of x? Ans. X= 44/12 A line parallel to the x-axis has a slope of zero while a line parallel to the y-axis has a slope of infinity. For perpendicular lines with slopes of m1 and m2, respectively, the slope pf one is the negative reciprocal of the other. Example 1 : Determine B such that 3x+2y – 7 = 0 is perpendicular to 2x – By + 2 =0. Ans. B=3

3. Angle Between Two Lines Consider two lines with slopes of m1 and m2. Example: Find the angle between the lines 3x+2y=6 and x+y =6 . ans. 11.3099 Distance Between a Point and a Line Example: Determine the distance from (5,10) to the line x-y=0. Ans. 3.54 units

4. Distance Between Two Parallel Lines Consider two parallel lines with equations as shown int the figure. The (perpendicular) distance, d, between the two lines is: Use the sign (either + or -) that would make the distance positive Example: Find the distance between the lines, 3x + y – 12 =0 and 3x + y – 4 = 0. ans. 2.53 units Division of Line Segment Consider two points with coordinates (X1,Y1) and (X2,Y2). The line segment formed by these two points is divided by a point P whose coordinates are (X,Y). Let r1 and r2 be the corresponding ratio of its length to the total distance between two points. Then: x= (x1r2)+(x2r1)/r1+r2 for y= (y1r2)+(y2r1)/r1+r2 Example: Determine the coordinates of the point which is 3/5 of the way from the point (2,-5) to the point (-3,5). )

5. Equations of Lines 1. General Equation: Ax+By+C =0 2. Points-Slope Form: y-y1=m(x-x1)

6. 3. Slope-Intercept Form: 5. Intercept Form: 4. Two-point Form: Example 1: Find the equation of a line where x-intercept is 2 and y-intercept is -2. ans. x-y-2=0 Example 2: What is the equation of the line joining the points (3, -2) and (-7, 6). Ans. 4x + 5y – 2 = 0

7. Area by Coordinates Example: Given three vertices of a triangle whose coordinates are A(1,1), B(3,-3) and C(5,-3). Find the area of the triangle. Ans. 4 square units Consider a polygon whose vertices have coordinates of (x1,y1), (x2,y2) and (x3,y3).

8. Exercises: is always measured from the y-axis. =___________________ is always measured from the x-axis. =___________________ Find the distance between point A(4,3) and B(2,5) =__________ A point P (X,3) is equidistant from the point (2,9) and (4,7). What is the value of x? =_________ Determine B such that 3x+2y – 7 = 0 is perpendicular to 2x – By + 2 =0. ____________________ Find the angle between the lines 3x+2y=6 and x+y =6 . =______________ Find the equation of a line where x-intercept is 2 and y-intercept is -2. =____________ _______________is regarded as the founder of analytic geometry by introducing coordinates system in 1637. If A = (-1, 5) and B = (7, -1) and AP/PB = 3, find the coordinates of P. =____________ Find the slope of the line passing the points (5, 0) and (4, 3). =_________________