Mathematics 1
Lecture 5
Pattarawit Polpinit
Lecture Objective
At the end of the lesson, the student is expected to be able to:!
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familiarize with the use of Cartesian Coordinate System.!
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determine the distance between two points.!
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define and determine the angle of inclinations and slopes
of a single line, parallel lines, perpendicular lines and
intersecting lines.!
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determine the coordinates of a point of division of a line
segment.
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Directed Line – a line in which one direction is chosen
as positive and the opposite direction as negative.!
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Directed Line Segment – consisting of any two points
and the part between them.!
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Directed Distance – the distance between two points
either positive or negative depending upon the
direction of the line.
Rectangular Coordinates
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A pair of number (x, y) in which x is the first and y being
the second number is called an ordered pair.!
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A vertical line and a horizontal line meeting at an origin,
O, are drawn which determines the coordinate axes.
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Coordinate Plane – is a plane determined by the
coordinate axes.
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X – axis – is usually drawn horizontally and is called as
the horizontal axis.!
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Y – axis – is drawn vertically and is called as the vertical
axis.!
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O – the origin!
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Coordinate – a number corresponds to a point in the
axis, which is defined in terms of the perpendicular
distance from the axes to the point.
Distance between two Points
1. Horizontal!
The length of a horizontal line segment is the abscissa (x
coordinate) of the point on the right minus the abscissa (x
coordinate) of the point on the left.
Distance between two Points
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Distance(d) = x2 - x1!
Distance between two Points
2.Vertical!
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The length of a vertical line segment is the ordinate (y
coordinate) of the upper point minus the ordinate (y
coordinate) of the lower point.
Distance between two Points
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Distance(d) = y2 - y1!
Slant
To determine the distance between two points of a slant
line segment add the square of the difference of the xcoordinates to the square of the difference of the yordinates and take the positive square root of the sum.
Slant
Sample Problems
1. Determine the distance between !
a. (-2, 3) and (5, 1)!
b. (6, -1) and (-4, -3)!
2. Show that points A (3, 8), B (-11, 3) and C (-8, -2) are vertices of an
isosceles triangle.!
3. Show that the triangle A (1, 4), B (10, 6) and C (2, 2) is a right
triangle.!
4. Find the point on the y-axis which is equidistant from A(-5, -2)
and B(3,2).
5. Find the distance between the points (4, -2) and (6, 5).!
6. By addition of line segments show whether the points
A(-3, 0), B(-1, -1) and C(5, -4) lie on a straight line. !
7. The vertices of the base of an isosceles triangle are (1,
2) and (4, -1). Find the coordinate of the third vertex if its
abscissa is 6.!
8. Show that the points A(-2, 6), B(5, 3), C(-1, -11) and D(-8,
-8) are the vertices of a rectangle. !
9. Find the point on the y-axis that is equidistant from (6,
1) and (-2, -3).
Inclination and Slope of a Line
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The inclination of the line, L, (not parallel to the x-axis)
is defined as the smallest positive angle measured from
the positive direction of the x-axis or the
counterclockwise direction to L.!
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!The slope of the line is defined as the tangent of the
angle of inclination.
Parallel and Perpendicular Lines
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Note for two lines with slopes m1 and m2:!
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If two lines are parallel, their slope are equal. !
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If two lines are perpendicular the slope of one of the
line is the negative reciprocal of the slope of the other
line.!
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or
Sign Conventions
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Slope is positive (+), if the line is leaning to the right.!
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Slope is negative (-), if the line is leaning to the left.!
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Slope is zero (0), if the line is horizontal.!
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Slope is undefined ( ), if the line is vertical.
Examples
1. Find the slope, m, and the angle of inclination, θ, of the
lines through each of the following pair of points.!
a.(-8, -4) and (5, 9)!
b.(10, -3) and (14, -7)!
c. (-9, 3) and (2, -4).!
2. The line segment drawn from (x, 3) to (4, 1) is
perpendicular to the segment drawn from (-5, -6) to (4,
1). Find the value of x.
3.Show that the triangle whose vertices are
B(5, -1) and C(-2,-8)! is a right triangle.! !
A(8, -4),
4.Show that the points A(-2, 6), B(5, 3), C(-1, -11) and D(-8,
-8) are the vertices of a parallelogram. Is the
parallelogram a rectangle?!
5.Find y if the slope of the line segment joining (3, -2) to
(4, y) is -3.!!
6.Show that the points A(-3, 0), B(-1, -1) and
on a straight line.
C(5, -4) lie
Find Angle between Lines
Find Angle between Lines
Examples
1.The line through the points (-1, 6) and (5, -2) intersects the
line through (4, -4) and (1,7). Determine the angle between
these two intersecting lines.!
2.The angle from the line through (x, -1) and (-3, -5) to the
line through (2, -5) and (4, 1) is 45 . Find x.! !
3.Two lines passing through (2, 3) make an angle of 45 . If the
slope of one of the lines is 2, find the slope of the other.!
4.Find the interior angles of the triangle whose vertices are A
(-3, -2), B (2, 5) and C (4,2).