Post on 03-Jun-2018
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MATH14
Plane and Solid Analytic Geometry
Fundamental Concepts
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SPECIFIC OBJECTIVES:
At the end of the lesson, the student is expected to beable 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.
Define the median of the triangle.
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FUNDAMENTAL CONCEPTS
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 coordinatesystem.
DEFINITION:
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Two Parts of Analytic Geometry
1. Plane Analytic Geometrydeals with figures
on a plane surface.2. Solid Analytic Geometrydeals with solid
figures.
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Directed Linea line in which one direction is chosenas positive and the opposite direction as negative.
Directed Line Segmentconsisting of any twopoints and the part between them.
Directed Distancethe distance between two pointseither positive or negative depending upon the
direction of the line.
DEFINITION:
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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.
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Coordinate Planeis a plane determined by thecoordinate axes.
o
y
x
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x
axisis usually drawn horizontally and is calledas the horizontal axis.
y axisis drawn vertically and is called as the
vertical axis.othe origin
coordinatea number corresponds to a point in
the axis, which is defined in terms of the
perpendicular distance from the axes to the point.
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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
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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.
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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.
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SAMPLE PROBLEMS
1. Determine the distance betweena. (-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 fromA(-5, -2) and B(3,2).
5. 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.
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6. The vertices of the base of an isosceles triangle are(1, 2) and (4, -1). Find the ordinate of the third vertexif its abscissa is 6.
7. Find the radius of a circle with center at (4, 1), if achord of length 4 is bisected at (7, 4).
8. Show that the points A(-2, 6), B(5, 3), C(-1, -11) andD(-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 thecoordinates of P.
10. Find the point on the y-axis that is equidistant from(6, 1) and (-2, -3).
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AREA OF A POLYGON BY COORDINATES
Consider 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
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Then the area of the triangle is determined by: [in
counterclockwise rotation]
1yx
1yx1yx
2
1A
33
22
11
Generalized formula for the area of polygon by
coordinates:
1n54321
1n54321
yy..yyyyy
xx..xxxxx
21A
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SAMPLE PROBLEMS
1. Find the area of the triangle whosevertices 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).