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