Day1. Introduction to the coordinate plane & distance formula

Teacher 1) introduces Rene Descartes who inveted the idea geometrically representing ordered pairs of numbers, and 2) explaines the maening the coordinate geometry between algebra and geometry. Teacher 3) teches the distance formula.

Coordinate Plane

Draw a straight horizontal line onthe graphing paper. As we count : "Zero, one, two, three.." we put the numbers on the line, one number per line of the graph paper. When we count backwards, we distinguish the numbers that come before zero by placing a "-" sign in front of them, so it goes: "Two, one, zero..." Make sure that you evenly space the numbers, since the distance from 1 to 2 should be the same as the distance from 2 to3.

What we have now is called a "number line" or "coordinate line." It can be used to describe where the point is on the line. To give the exact "address" of a point, we just look at how far the point is from zero, using a minus sign for numbers to the left of zero.

To get more freedom of movement, we will let our points be anywhere on the paper, not only on the line. We can still count from a selected point or zero, but now we will need more than just one number to give the exact "address" of a point. For example, can you describe how to get to point A from zero? We can think about the grid as streets, and squares as blocks, so we are only allowed to go by the grid lines.

There are many ways to get from point to point (how many ways, by the way?). To create a standard way of referring to points, scientists came to an agreement that they will always name the point after one special way of walking. As usual, starting from zero, we go all the way to the right or to the left, counting steps: one, two. Then we go up or down: one, two three steps up. Then wewrite the number of steps like that: (2,3). Again, the first number is "left-right," the second "up-down." A minus sign means either left or down. So, if our point is
(-2, -3), we go two steps to the left, and then three steps down.

History of the Coordinate Geometry

Rene Descartes, who was born in 1596 invented the idea o geometrically representing ordered pairs of numbers. He was thrilled with his invention, which he called a method, for it used algebra to combine arithmetic and geometry, ad so unified all the mathematics known up to that time. He used his method, which is now called coordinate geometry or analytic geometry, to solve many problems which were then very difficult or not to be solved.

(Sometimes a distinction is made between analytic geometry and coordinate geometry. Analytic geometry refers to the use of graphs to describe equations and inequalities as well as to the study of geometric figures. Coordinate geometry refers to the use of the coordinate plane to study geometric figures. That is, in coordinate geometry the figure comes first and it is placed on a coordinate plane for study. In anlytic geonetry, and equation often comes first, and the geometry is used to study the equation.)

Descartes believed that he had found a method whereby any mathematics problem could be solved or any conjecture proved or disproved. In fact, he thought that mathematics and logic could provide the means whereby any problem in any field of endeavor could be solved. Today we know that his belief is not even theoretically possible, but Descartes' dream reflects the power of the coordinate geometry methods he discovered.

A figure in many ways

Figures can be described with or without coordinates. Above and below are three descriptions of congruent rectangular regions. Two of these descriptions use coordinates. Any polygon or polynonal region can be described using a description like the one just below on the left.

Distance Formula using the PythagoreanTheorem

Let P=and R=. First find Q so that is the hypotenuse of a right triangle. Such a point is Q=.

Since a number and its absolute value have the same square,

and .

Thus

.

Taking square roots

.

This gives a formula we should memorize.

Distance formula The distance between two points and in the coordinate plane is

Example

1) Find the distance between (-5,7) and (10,-2).

2) To get to a hospital from the middle of a nearby town, you can drive 8 miles east, turn and go 4 miles south, and then go 1 miles west. By helicopter, how far is it from the middle of the town to the hospital?

3) Let A=(-5, 0), B=(5,8), and C=(4, -1). Prove that triangle ABC is isosceles by using the distance formula.

Day 2. Equation for Circles

At the first day, the coordinate plane and the distance formula are introduced. Many students are familiar with the equation of the lline. Now we will study the equation of the circle using the coordinate plane.

Equations for circles

Here is the circle with center (3,2) and radius 10. By adding or subtracting 10 from either coordinate of (3,2), four points on the circle can be found. there are (13,2), (3,12), (-7,2), and (3,-8). It would be nice to find an equation satisfied by these four points and all pther points on the circle.

If a point (x,y) is on the circle, then by the Distance Formula,

=10

This equation is an equation for the circle. However, most people prefer equations without square roots. Squaring both sides give an equivalent equation.

=100

To check if this is correct, try the point (-7,2). It should satisfy the equation. Substitute -7 for x and 2 for y.

It is easy to generalize this example.

Equation of a cicle The circle with center (h, k) and radius r is the set of points (x,y) satisfying

Proof

Since r is the radius, by the definition of circle, the distance from (h, k) to any point (x, y) on the circle is r.

That distance is given by the Distance Formula.

Squaring both sides:

A major difficulty some students have in understanding the equation for a circle is that the use of the variables x and y in the formula is different from the use of h, k, and r. The variables x and y may stand for any point on the circle, but h, k, and r are constants. Teacher might point out that the expressiogives the distance from any point (x, y) to (3, 2). By setting the expression equal to 10, only those points 10 units away from (3, 2) will satisfy the equation. The constants are 3, 2, and 10.

Example

1) Write an equation for the circle with center (-5, 0) and radius 6.

2) Find the center and radius of the circle with equation (x+3) +(y-2) = 16.

3) A circle has center (2, -1) and touches te x-axis at exactly one point. The circle is called tangent to the axis.

a. Draw a picture.

b. What are the coordinate of the point of tangency?

c. Find an equation for the circle.

d. Find the area of this circle.

Day 3 The midpoint formula

Suppose we are goint to find the center of the gravity of a book. The book can be coordinated as a set of points in the coordinate plane : (-3, 6), (7, 2), (2, -2), and (-4, 1). The coordinate of the center of gravity are found by computing the mean of the x-coordinates and the mean of the y-coordinates of the points. So the mean of the x-coordinates is (-3+7+3+ -2)/4=2/4= 0.5, and the mean of the y-coordinates is (6+2+ -2+1)/4=7/4=1.75. So the center of gravity is (0.5, 1.75), as shown below.

In the simplest case, there are only two points to begin with, and the center of gravity is th midpoint of the segment connecting them.

Midpoint Formula If a segment has endpoint (a, b) and (c, d), its midpoint is

Proof

Draw a picture with P=(a,b), Q=(c, d), and M=.

Recall the definition of midpoint. To show that M is the midpoint of , we need to show that PM=MQ and M is on .

The algebra that follows is cumbersome, but it works out. To show that PM=MQ, we calculate ditances using the Distance Formula.

Thus PM=MQ. To show that M is on , we calculate the slopes of and .

slope of

slope of

The slopes are equal so //. Both lines contain point M, so = and m is on . So M is the midpoint of .

Example

1) If P=(-10, 6) and Q= (1, 8), find the midpoint of PQ.

2) Let A=(0,12), B=(2,-4), and C=(8,10), as shown below. Let M and N be the midpoints of AB and AC. Prove MN//BC.

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