# Distance, Midpoint &, Circles in the Coordinate Plane

** Graphs and Equations **

Start by making statements such as the fol lowing : Graphs provide a means
of displaying, in-

terpreting, and analyzing data in a visual format. Many real -world situations
can be described

mathematically using equations in two variables . Each pair of variables
re presents a solution to

the equation and can be displayed as a single point, or ordered pair (x, y), on
a coordinate plane.

The coordinate plane is devised of two perpendicular lines called the
which divide

the plane into four regions, called . The
horizontal line is called the

and the vertical line is called the .

** Plot points :** *At this point, I would ask students
how to describe how to plot a point or two, which*

quadrant the point lies in, etc.

**Solutions of Equations:** *How many different ways
can you show, or prove, that a pair is a*

solution? The relationship among ordered pairs, solutions to equations, and
points on a graph is

essential.

De termine whether each ordered pair is a solution of 2x + 3y = 18.

(a) (-5, 7)

(b) (3, 4)

(c) If (4,A) is on the graph of 2x - 5y = 8, find A. *This is a typical
homework/test question.*

**Graphs of Equations:**

To graph an equation is to make a drawing that represents the
of that equation.

*Students should be able to graph by plotting points, key points are the
intercepts.*

An x-intercept is a point (a, 0). To nd a,
.

The x-intercept is also called a zero because

A y-intercept is a point (0, b). to nd b, **
.**

Examples: Graph by nding intercepts.

2x + 3y = 18

3x - 5y = -10

Graph y = x^{2}-9x-12 by plotting points. [Create a table of points by using a
graphing calculator.]

__Distance __*This is one of the last topics in Math
002 and should be familiar to students.*

The distance d(P_{1}, P_{2}) between two points P_{1}(x_{1}, y_{1}) and P_{2}(x_{2}, y_{2}) is
given by

Use the distance formula to verify that (-6, 3), (3,-5),
and (-1, 5) form a right triangle. .. *or*

some other example where the students need to show they know how to use the
distance formula.

Find all points having an x-coordinate of 4 whose distance
from (-3,-1) is 13. *This can be done*

by the distance formula and/or by plotting points and reasoning through the
Pythagorean Theorem.

Start by making a sketch, then ask for suggestions.

__Midpoint__ *Midpoint is pretty intuitive, although
students may later confuse the formula with the*

formula for slope.

The coordinates of the midpoint of a line segment with endpoints (x_{1}, y_{1}) and
(x_{2}, y_{2}) is given by

Example: Find the midpoint of the hypotenuse in the right
triangle above. *Extra. Use if time and*

if the above example was used.

__Circles__ *Circles will be new to Math 002
students. Many have not had a geometry course for several*

years. Connect to the distance formula.

De nition: A circle is the set of all points that are
from a given point. The

given point is . The constant distance is
called the .

The general form for the equation of a circle is:

The standard form for the equation of a circle is:
where

is the
and is the
.

1. Identify the center and radius for the circle given by
(x - 1)^{2} + (y - 5)^{2} = 4.

2. Write the equation for the circle with center at
and radius
. Sketch the graph.

*Do variations of this question with center on an axis, in di erent quadrants,
etc. and radii*

that are integers as well as simple radicals , like
and so on.

3. Find the center and radius of the circle given by x^{2} +
y^{2} - 6x - 4y - 4 = 0. *completing the*

square is a familiar , but not necessarily pro cient, skill for Math 002
students.

4. What is the center of the circle with diameter
endpoints at (-2, 5) and (4,-3)? Write the

equation of this circle.