Mathematics 1. Lecture 5. Pattarawit Polpinit

Transcription

1 Mathematics 1 Lecture 5 Pattarawit Polpinit

2 Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance between two points. define and determine the angle of inclinations and slopes of a single line, parallel lines, perpendicular lines and intersecting lines. determine the coordinates of a point of division of a line segment.

3 Directed Line a line in which one direction is chosen as positive and the opposite direction as negative. Directed Line Segment consisting of any two points and the part between them. Directed Distance the distance between two points either positive or negative depending upon the direction of the line.

4 Rectangular Coordinates A pair of number (x, y) in which x is the first and y being the second number is called an ordered pair. A vertical line and a horizontal line meeting at an origin, O, are drawn which determines the coordinate axes.

5 Coordinate Plane is a plane determined by the coordinate axes.

6 X axis is usually drawn horizontally and is called as the horizontal axis. Y axis is drawn vertically and is called as the vertical axis. O the origin 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.

7 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.

8 Distance between two Points Distance(d) = x 2 - x 1

9 Distance between two Points 2.Vertical 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.

10 Distance between two Points Distance(d) = y 2 - y 1

11 Slant To determine the distance between two points of a slant line segment add the square of the difference of the x- coordinates to the square of the difference of the y- ordinates and take the positive square root of the sum.

12 Slant

13 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).

14 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 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).

15 Inclination and Slope of a Line 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. The slope of the line is defined as the tangent of the angle of inclination.

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18 Parallel and Perpendicular Lines Note for two lines with slopes m 1 and m 2 : If two lines are parallel, their slope are equal. If two lines are perpendicular the slope of one of the line is the negative reciprocal of the slope of the other line. or

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20 Sign Conventions Slope is positive (+), if the line is leaning to the right. Slope is negative (-), if the line is leaning to the left. Slope is zero (0), if the line is horizontal. Slope is undefined ( ), if the line is vertical.

21 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.

22 3.Show that the triangle whose vertices are A(8, -4), B(5, -1) and C(-2,-8) is a right triangle. 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 Show that the points A(-3, 0), B(-1, -1) and on a straight line. C(5, -4) lie

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24 Find Angle between Lines

25 Find Angle between Lines

26 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 line through (2, -5) and (4, 1) is 45. Find x. (-3, -5) to the 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).