Analytic Geometry (Lesson 1)
Analytic Geometry (Lesson 1)
Analytic Geometry (Lesson 1)
Math 14
OBJECTIVES:
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. Determine the area of a polygon by coordinates.
FUNDAMENTAL CONCEPTS
DEFINITION: Analytic Geometry is the branch of mathematics, which deals with the properties, behaviours, and solution of points, lines, curves, angles, surfaces and solids by means of algebraic methods in relation to a coordinate system.
DEFINITION:
Directed Line a line in which one direction is
chosen as positive and the opposite direction as negative.
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.
P (x, y)
2. Vertical The length of a vertical line segment is the ordinate (y-coordinate) of the upper point minus the ordinate (ycoordinate) of the lower point.
3. Slant To determine the distance between two points of a slant line segment add the square of the difference of the abscissa to the square of the difference of the ordinates and take the positive square root of the sum.
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. By addition of line segments show whether the points A(-3, 0), B(-1, -1) and C(5, -4) lie on a straight line. 6. The vertices of the base of an isosceles triangle are (1, 2) and (4, -1). Find the ordinate of the third vertex if its abscissa is 6. 7. Find the radius of a circle with center at (4, 1), if a chord of length 4 is bisected at (7, 4). 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. The ordinate of a point P is twice the abscissa. This point is equidistant from (-3, 1) and (8, -2). Find the coordinates of P. 10. Find the point on the y-axis that is equidistant from (6, 1) and (-2, -3).
P1 x1 , y1
P3 x3 , y3
P2 x2 , y2
Then the area of the triangle is determined by: [in counterclockwise rotation]
1 A x2 2 x3 x1 y1 y2 y3 1 1 1
1 x1 A 2 y1
x2 y2
x3 y3
x4 y4
x5 y5
. . xn . . yn
x1 y1
SAMPLE PROBLEMS
1. Find the area of the triangle whose vertices are (-6, -4), (-1, 3) and (5, -3). 2. Find the area of a polygon whose vertices are (6, -3), (3, 4), (-6, -2), (0, 5) and (-8, 1). 3. Find the area of a polygon whose vertices are (2, -3), (6, -5), (-4, -2) and (4, 0).
REFERENCES
Analytic Geometry, 6th Edition, by Douglas F. Riddle Analytic Geometry, 7th Edition, by Gordon Fuller/Dalton Tarwater Analytic Geometry, by Quirino and Mijares