Foundation,
Distance,
and Midpoint
Formula Formula Notes
Module
1 - Foundation
of Geometry
Module1 Lesson
7 Lesson
1: Foundation,
Distance,
and Midpoint
A definition uses known words to describe a new word.
In geometry, some words such as point, line, and plane are undefined terms. There is no formal definition for
these words, but instead they are explained by using examples and descriptions which allows us to define
other geometric terms and properties.
The 3 Undefined Terms of Euclidean Geometry
‡
Point '&
A point is simply a location. It has no dimension (shape or size), is usually represented by a small dot,
and named by a capital letter.
!"
x
!"#$%&'&
Line
A line is a set of points and extends in both directions. It is one-dimensional. It has no thickness or
width.
It is usually represented by a straight line with two arrowheads to indicate that it extends without end in
both directions, and is named by a lowercase script letter or any two points on the line.
x
Plane
A plane is a flat surface made up of points. It is two dimensional, is usually represented by a shape that
looks like a tabletop or wall, and is named by a capital script letter or 3 (or 4) noncollinear points. You
must imagine that the plane extends without end, even though the drawing of a plane appears to have
edges.
!"
Plane !"or Plane ABC
!"
#"
$"
Definitions
x
Space: boundless three-dimensional set of all points. It can contain points, lines and planes
x
Collinear Points: points that lie on the same line. Any two points are collinear.
x
Noncollinear Points: points that do not lie on the same line. In the above plane, points A, B and C are
noncollinear.
x
Coplanar Points: points that lie in the same plane. Any 3 noncollinear points are coplanar.
x
Noncoplanar Points: points that do not lie in the same plane.
Examples:
D
1) Name the line shown in the picture.
!"
There are several names:
#"
!"
$"
E
DA , DE or AE (Also AD ,
ED , or EA )
2) Name 3 collinear points.
Points D, A, and E are collinear since they are on the
same line.
3) Name 3 coplanar points.
Points A, B, C are coplanar and that plane is shown in
the drawing (plane M). Another set of coplanar points
would be D, B, A. They are noncollinear so they meet
the qualification for being a plane. The plane containing
those points is just not drawn in this figure. Can you
determine any other sets of coplanar points?
4) Name 4 noncoplanar points.
A, B, C, D or
(Notes continue ± keep scrolling down)
A, B, C, E
Distance and Midpoint
You are familiar with the coordinate plane from Algebra I. In coordinate geometry you use an ordered pair (x, y)
to describe the location of a point on the Euclidean coordinate plane.
You can measure the distance between any two points if the points are on a vertical line or a horizontal line. If
the two points do not line up vertically or horizontally, you can use the Distance Formula to find the distance
between two points.
The Distance Formula
The distance d between any two points P(x1, y1) and Q(x2, y2) is given by
d=
Example: Find the distance between J(-4, 1) and T(1, -2) from the illustration in the graph above.
d = JT =
=
=
=
.
1RZ³You 7U\´these problems (answers are located at the end of this document so you can check them.
0HVVDJH\RXUWHDFKHULI\RXFDQ¶WJHWWKHULJKWDQVZHUVDQGQHHGKHOS
1) Find the distance between N(-4, 0) and W(3, 6).
2) Find the distance between S(0, -1) and B(5, -2).
M idpoints
If you were to pick two points on the number line, say 2 and 10, you know that the segment has a length of 8
units. You already know that the midpoint bisects a segment. So the midpoint of the segment between 2 and 10
has a length of 4 units.
You find the midpoint by taking the average of the values of the two endpoints of the segments:
2
= 12 = 6 = midpoint
2
When you work in the coordinate plane, you find the midpoint for the x-values and the midpoint for the yvalues. This result is another point.
The Midpoint Formula
The coordinates of the point M of
with endpoints P(x1, y1) and Q(x2, y2) are
M = ( x1 + x2 , y1 + y2 )
2
2
Now, ³<RX7U\´ WKHVHSUREOHPV'RQ¶WIRUJHWWRFKHFN\RXUDQVZHUVDWWKHHQGRIWKLVGRFXPHQW
3) Find the midpoint between N(-4, 0) and W(3, 6).
4) Find the midpoint between S(0, -1) and B(5, -2).
5) If the midpoint is C(-2, 3) and one endpoint is at A(4, 2), where is the other endpoint B?
Answers to Distance and Midpoint ³<RX7U\´problems:
1) NW =sqrt[
2
2
2) SB = sqrt[(5-0)2 + (-2 ± (-1))2 ] =
3) midpoint = (
4) midpoint = (
5) ± 2 = 4 + x
2
±2=4+x
1
2
-4 = 4 + x
-8 = x
and
3=2+y
2
3=2+y
1
2
6=2+y
4=y
]=
=
=
)=(
)=(
)
cross multiply here!
cross multiply!
FINAL ANSWER B is (-8, 4)
=
=