analytic geometry of lines

2. WHAT IS A LINE?
• From geometry, we already know that a line is determined by two
distinct points; and that it is composed of infinitely many points.
• Since a line is a set of points, technically, it is also a locus.
• A line is a locus of points that have a constant slope.
• A line intersects the x and y axes at the points which are called the x-
intercept and y-intercept, respectively.

3. PROPERTIES OF A LINE
• constant slope (m)
• infinitely many points. • x - intercept
• y - intercept
y - intercept
x - intercept
constant slope
constant slope
constant slope
y - intercept
x - intercept

4. SLOPE OF A LINE
• The slope of a line is determined by the ratio of the difference of the y-
coordinates to the x-coordinates of two points on the line.
• It is represented as m and describes the steepness of the line it
represents.
m = 0
m > 0 m < 0
m = undefined
• also tells if lines are parallel, perpendicular, or simply intersects.
slopes are
equal
slopes are
negative
reciprocals
slopes are not equal nor
negative reciprocals

5. • An intercept of a line is the point that the line intersects with the x or y
axis.
• The point on the x-axis that a line intersects is the x-intercept while the
point on the y-axis that it intersects is the y-intercept.
• An x-intercept is written as (a, 0) while a y-intercept is written as
(0, b), for any real number a and b.
• A vertical line does not have a y-intercept, while a horizontal line does
not have an x-intercept.
X AND Y INTERCEPTS

6. • A line is represented by an equation of the form Ax + By + C = 0, where
A, B, and C are integers.
• The properties of a line such as the slope, intercepts, or any pair of
points can create the equation of a line.
• There are four main forms of the equation of a line, all of which are
based on the mentioned properties.
EQUATION OF A LINE
Two-point
Form
Point-slope
Form
Slope-intercept
FormIntercepts
Form

7. TWO-POINT FORM
**Should be written in Ax + By + C = 0 form.

8. POINT-SLOPE FORM

9. INTERCEPTS FORM
NOTE: Two-point form can be applied if given the two intercepts.

10. SLOPE - INTERCEPT FORM

11. USING THE SLOPE-INTERCEPT FORM GIVEN
SLOPE BUT NOT Y-INTERCEPT
We don’t have b, but
we know values for x
and y.
Solve for
b, then
substitute it
back to the
equation.

12. GRAPH OF A LINE
• Just plot the points.
• Connect the points with a straight line.
• This is also the approach when the x
and y intercepts are given.

13. GRAPH OF A LINE
• Just plot the point.
• Get a second point by using the slope.
(in this example, the slope tells us that
another point is two units down and
three units to the right of the given point)
•Connect the points with a straight line.
• This is also done when the slope and
an x or y intercept is given.

14. GRAPH OF A LINE

15. SOME NOT-SO-BASIC SITUATIONS
INVOLVING LINES
• Remember how slopes of parallel and perpendicular lines are related.
This is used in determining equations of lines through a point given a line
parallel or perpendicular to it.
PARALLEL AND PERPENDICULAR LINES
slopes are
equal
slopes are
negative
reciprocals

16. SOME NOT-SO-BASIC SITUATIONS
INVOLVING LINES
The slope of the given
line will be the slope of
the line being asked for
because they are
parallel.
Since we have a
slope and a point for
the line being asked
for, we can use
point-slope form.

17. SOME NOT-SO-BASIC SITUATIONS
INVOLVING LINES
The negative reciprocal of
the slope of the given line
will be the slope of the
line being asked for
because they are
perpendicular.
Since we have a
slope and a y-
intercept for the line
being asked for, we
can use slope-
intercept form.

18. LINE FAMILIES