Math14 lesson 1
-
Upload
warren-cunanan -
Category
Technology
-
view
5.028 -
download
2
Transcript of Math14 lesson 1
![Page 1: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/1.jpg)
ANALYTIC GEOMETRY(Lesson 1)
Math 14 Plane and Analytic Geometry
![Page 2: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/2.jpg)
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.
![Page 3: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/3.jpg)
• 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:
FUNDAMENTAL CONCEPTS
![Page 4: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/4.jpg)
Two Parts of Analytic Geometry
1. Plane Analytic Geometry – deals with figures on a plane surface.
2. Solid Analytic Geometry – deals with solid figures.
![Page 5: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/5.jpg)
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.
DEFINITION:
![Page 6: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/6.jpg)
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.
![Page 7: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/7.jpg)
Coordinate Plane – is a plane determined by the coordinate axes.
o
y
x
P (x, y)
![Page 8: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/8.jpg)
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 origincoordinate – a number corresponds to a point in the axis, which is defined in terms of the perpendicular distance from the axes to the point.abscissa – is the x-coordinate of an ordered pair.ordinate – is the y-coordinate of an ordered pair.
![Page 9: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/9.jpg)
DISTANCE BETWEEN TWO POINTS
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.
1. Horizontal
![Page 10: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/10.jpg)
![Page 11: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/11.jpg)
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.
![Page 12: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/12.jpg)
![Page 13: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/13.jpg)
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.
![Page 14: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/14.jpg)
![Page 15: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/15.jpg)
SAMPLE PROBLEMS1. 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.•Show that the triangle A (1, 4), B (10, 6) and C (2, 2) is a right triangle.•Find the point on the y-axis which is equidistant from A(-5, -2) and B(3,2).
![Page 16: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/16.jpg)
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).
![Page 17: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/17.jpg)
AREA OF A POLYGON BY COORDINATESConsider the triangle whose vertices are P1(x1, y1), P2(x2, y2) and P3(x3, y3) as shown below.
o
y
x
111 y,xP
222 y,xP
333 y,xP
![Page 18: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/18.jpg)
Then the area of the triangle is determined by: [in counterclockwise rotation]
1yx
1yx
1yx
2
1A
33
22
11
Generalized formula for the area of polygon by coordinates:
1n54321
1n54321
yy..yyyyy
xx..xxxxx
2
1A
![Page 19: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/19.jpg)
SAMPLE PROBLEMS1. 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).
![Page 20: Math14 lesson 1](https://reader036.fdocuments.us/reader036/viewer/2022062319/555dc595d8b42aec698b4988/html5/thumbnails/20.jpg)
REFERENCES
Analytic Geometry, 6th Edition, by Douglas F. RiddleAnalytic Geometry, 7th Edition, by Gordon Fuller/Dalton
TarwaterAnalytic Geometry, by Quirino and Mijares