lecture #2 analytic geometry
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ANALYTIC GEOMETRYMATH 004
Lecture #2• Slope of a Line• Slope of a line in terms of angle of
inclination• Slope of a line in terms of coordinates• Slope of Parallel line• Slope of Perpendicular line• Angle formed between two lines
SLOPE OF A LINEThe slope or gradient of a line describes the
steepness, incline or grade.A higher slope value indicates a steeper
incline.Slope is normally described by the ratio of the
“rise” divided by the “run” between two points on a line.
The slope is denoted by .
SLOPE OF A LINEIf line rises from left to right,
SLOPE OF A LINE• If line goes from right to left
SLOPE OF A LINE• If line is parallel to x-axis
SLOPE OF A LINE• If line is parallel to y-axis
SLOPE OF A LINE IN TERMS OF ANGLE OF INCLINATION
The slope of a line is the tangent of the inclination, thus
Prove!1.The slope of a vertical line is undefined!2. The slope of a horizontal line is zero!
SLOPE OF A LINE IN TERMS OF COORDINATES
The slope of a line passes through points and is
SLOPE OF PARALLEL LINESTwo non-vertical lines and are parallel if and
only if their slopes are equal and the angle of inclination are also equal.
(slope)
(angle ofinclination)
SLOPE OF PERPENDICULAR LINES
Two non-vertical lines and are perpendicular if and only if their slopes are negative reciprocals of each other.
or
or
ANGLE BETWEEN TWO LINES
The between two lines and is the angle formed by the rotating counter clockwise until it coincides with and it is given by
Sample Problem1.Find the slope and the angle of inclination of the line joining and .
2. A line with an angle of inclination of passes through the point . If the ordinate of the point on the line is , what is its abscissa?
3. Show by means of slope that the points , , and are the vertices of a parallelogram.
4. Show by means of slope that the points , and are vertices of a right triangle.
5. Find the angle from the line through the points and to the line through the points and .
6. The angle between two lines and is . If the gradient of is , find the gradient of .