Day 1 Distance of a segment
objectives:
[1] to define abscissa and ordinate
[2] to find the length of a line segment
In coordinate geometry, familiar geometric ideas are expressed in terms of numbers.Each point of a plane is associated with an ordered pair of real numbers, called coordinates of a point. Also, each ordered pair of real numbers is associated with a point of the plane. This idea is illustrated in the diagram.
As illustrated in the diagram, the x- and y-axes separate the plane into four regions, called quadrants, which are numbered I, II, III, IV, as shown. The x-coordinates are known as the abscissa and the y-coordinates are known as ordinates.
In the first section, we will see how numbers can be used to express the distance between two points and describe the midpoint of a line segment.
The distance between two points A and B will be denoted by AB. . To find the distance AB, horizontal and vertical lines are drawn through A and B to form a right triangle. The horizontal distance AC is found by calculating the absolute value of the difference of the x-coordinates of C and A. The vertical distance BC is found by calculating the absolute value of the difference of the y-coordinates of B and C. Then the Pythagorean Theorem is used.
and The use of the Pythagorean Theorem yields:
for the figure above. illustrates the same idea for two arbitrary points, [X1,Y1] and [Y1,Y2].This equation is known as the distance formula; it is a tool which allows you to calculate and compare distances. You may find it easier to remember this formula by visualizing the diagrams above and using the Pythagorean Theorem.
Day 2 Midpoint of a segment
objectives:
[1] to find the midpoint of a line segment
What are the coordinates of the midpoint M on the segment AB shown in figure [a] below?
figure [a] figure [b]
Figure [b] shows that the x-coordinate of M is 3=[1+5]/2, the average of 1 and 5. Likewise, figure [c] shows that
the y-coordinate of M is 4=[2+6]/2,
the average of 2 and 6.
A generalization of this reasoning is shown below. In the diagrams the x- and y-coordinates of the midpoint are denoted by x and y.
Summary
In coordinate geometry, familiar geometric ideas are expressed in terms of numbers. A point in the
plane is identified with a pair of numbers called its coordinates. Moreover, if A=[X1,Y1] and B=[X2,Y2] are two points of the plane, then the length and midpoint of AB can be expressed in terms of numbers by the following formulas:
Distance formula:
Midpoint formula:
Day 3 Linear Equations
objectives:
[1] to use algebraic equations to describe lines
A solution of the equation 2x + 3y = 12 is an ordered pair of numbers [x , y] which makes the equation true. For
example, since x = 0 and y = 4 make the equation true, the pair [0 , 4] is a solution. Another solution is [-3 , 6]
because 2[-3] + 3[6] = 12. The set of all points in the plane corresponding to ordered-pair solutions of the
equations is called the graph of the equation.
The equation 2x + 3y = 12, whose graph we have just considered, is one of the simplest kinds of equations in two
variables. The graph of any equation having the form ax + by = c is often
referred to as the line "ax + by = c."
There are many different linear equations which have the same graph. All these equations are related to each other by simple algebraic transformations. For example, the equations 4x - 2y = 2, 2x - y = 1, and y = 2x -1 all have the same graph. Of these three equations, the last one is most useful for making a table of points for the graph.
| x | y = 2x - 1 |
| 0 | -1 |
| 1 | 1 |
| 2 | 3 |
| 3 | 5 |
If a = 0 or b = 0, the equation ax + by = c looks strange. Consider for example, the equation y = 4. This looks like a
statement that some number y equals 4. But it really refers to the set of ordered pairs [x ,y] for which y = 4. Its
graph is shown below. Likewise, the graph of x = 2,
also shown below, consists of all ordered pairs [x , y ] for
which x = 2.
You can find the intersection of two lines
geometrically by plotting their graphs. In this section you will see how
the intersection can be found algebraically. Consider the following pair of equations:
3x + 2y = 12
-4x + 5y = 15
The graphs of both equations are straight lines which intersect in a point. To find this point, we solve the equations
simultaneously. One method of doing this is to multiply both sides of the first equation by 4 and both sides of the
second by 3, obtaining the following equations:
3x + 2y = 12
-4x + 5y = 15
y = 93/23
In a similar manner, we find that x = 30/23. Thus, [30/23 , 93/23] is the common solution of the two linear
equations. It is also the point common to their graphs.
However, two lines in a plane do not have to intersect. For example, the equations
6x + 4y = 8
3x + 2y = 1
have parallel line graphs. If you try to solve these equations simultaneously, you will find that they contradict
each other and therefore have no common solution.
A slight modification of the previous two equations gives us a different geometric picture. Consider the following
linear equations:
6x + 4y = 8
3x + 2y = 4
These equations have an infinite number of common solutions, and their graphs are the same line.
Day 4 The Slope of Lines
objectives:
[1] to find the slope of lines
In geometry we can prove that two lines L1 and L2, are parallel by proving that a pair of corresponding angles are congruent. It is a weakness of coordinate geometry that these angles cannot be evaluated easily. However, there is a concept in coordinate geometry that describes the steepness of a line relative to the x-axis. It is called slope, and we shall see that the lines L1 and L2 can be proved parallel by showing that they have the same slope.
The slope of a nonvertical line through the points [X1,Y1] and [X2,Y2] is defined by the equation
Thus, to determine the slope of a line, choose any two points on the line, for instance, [2,0] and [5,6] in figure [a] below. Subtract the y-coordinates, 6-0=6, and also the x-coordinates, 5-2=3. The ratio of these numbers, 6/3 = 2, is the slope. Notice that if you consider two other points, [-1,-6] and [0,-4], you get the same value for the slope.
Now let us imagine the line in figure [a] rotating counterclockwise about the point [2,0]. In figure [b], you see that as the line becomes steeper, its slope gets very large. In fact, when the line is vertical, as in figure [c], it is so steep that some people say that the line has infinite slope, although it is really more precise to say that it has no slope because the denominator is zero.
If you imagine the line rotating still more about [2,0], you will see that the slope becomes negative. [See figure [d] and [e].] The horizontal line through [2,0] has slope zero. [See figure [f].] Note that "zero slope" and "no slope" mean different things, as the diagram illustrates.
The following theorem proves that the slope [Y2-Y1]/[X2-X1] is the same for any two points of a line. It also provides an easy way way to find the slope of a line from its equation.
Theorem: The slope of the line with equation y = mx + k is m.
Proof: Let P = [X1,Y1] and Q = [X2,Y2] be any two different points of the line with equation y = mx + k. Then Y1 = mx + k and Y2 = mx + k. Therefore, the slope equals
[Y2-Y1] / [X2-X1] = [mX2 + k] - [mX1 + k] / [X2 -X1] =
m[X2 - X1] / [X2 - X1] = m.
When the equation of a line is written in the form y = mx + k, the numbers m and k should provide you with a mental picture of the line. According to the theorem, m is the slope of the line. The number k is called the y-intercept of the line because the line intersects the y-axis at the point [0,k]. The diagrams below illustrate the effect of m and k on the graph of the equation y = mx + k. Notice in the diagram that lines with the same slope are parallel.
Given the equation 7x + 13y = 26, find the slope and y-intercept of the line.
Solution: First rewrite the equation as follows:
7x + 13y = 26
13y = -7x + 26
y = [-7/13]x + 2
The slope is then -7/13, the coefficient of x. The y-intercept is 2.
In general, if the equation of a line is written in the form
ax + by = c, then the slope m of the line is -a/b and its y-intercept k is c/b.
Day 5: Parallel and Perpendicular Lines
objective:
[1] to show how the slope is related to the geometric ideas of parallel and perpendicular lines
In our development of coordinate geometry, we have shown how algebraic relationships involving numbers are used to describe geometric relationships involving points and lines.
| Algebraic Idea | Geometric Idea |
| An ordered pair of numbers [x,y] | A point in the plane |
| An equation ax + by = c [a and b not both 0] | A line in the plane |
| Solution of two linear equations | Point common to two lines |
| The number [Y2-Y1]/[X2-X1] | Slope of the line joining two points [X1,Y1] and [X2,Y2] |
| The number | Distance between points [X1,Y1] and [X2,Y2] |
| An ordered pair of numbers | Midpoint of segment joining points [X1,Y1] and [X2,Y2] |
We shall extend these lists by proving two theorems which show how the slope is related to the geometric ideas of parallel and perpendicular lines.
Theorem1:
a. If two nonvertical lines are parallel, then they have the same slope.
b. If two [different] nonvertical lines have the same slope, then they are parallel.
Proof: Suppose L1 and L2 are nonvertical lines as shown. Note that the slope of L1 is BC/AC and the slope of L2 is EF/DF.
A. 1. Suppose L1 is parallel to L2.
2. Then