Name:
Math 2
Date:
Coordinate Proofs
Objective:
Prove geometric theorems using coordinate methods.
Introduction to Coordinate P
roofs
Over the past few lessons, we have studied how to perform all the basic kinds of coordinate
calculations: finding slopes, distances, midpoints
, and points forming other ratios. While
applying these methods to particular diagrams, we often saw outcomes such as slopes turning out
to be equal, distances turning out to be equal or in a particular ratio, and midpoints of segments
turning out to be th
e same point.
These occurrences were often an indication that there is a geometric theorem (a provable fact
about geometry) that applied to that type of diagram. We are now turning our attention to
identifying these theorems and writing proofs of them, usi
ng the machinery of coordinate
calculations.
(Working with coordinates is just one of the possible approaches to geometric proof. We will
pursue other proof techniques in the next unit of the course: Unit 4, Deductive Geometry.)
The idea of a
coordinate p
roof
is to verify a geometric theorem using the relevant coordinate
calculations. For example:
Facts about equal lengths and other length relationships can be proved by calculating lengths
using the distance formula
and comparing th
em.
Facts about parallel lines or perpendicular lines can be proved by calculating slopes using
and comparing them (equal slopes indicate parallel; slopes having a product of
–
1
indicate perpendicular).
Facts concerning midpoints can
be proved by using the midpoint coordinates
in appropriate distance or slope calculations.
Facts concerning points forming other ratios can be proved using the formulas we devised at our
last class. For example, the point that’s
of the way from (
x
1
,
y
1
) to (
x
2
,
y
2
) can
be calculated as
(
x
1
+
x
2
,
y
1
+
y
2
). For other fractions, replace
and
with any
k
and (1
–
k
).
Usually we will be trying to prove a theorem about all shapes of a particular kind (examples:
about
all triangles, or about all right triangles). An important strategy for writing coordinate
proofs is to work with a s
hape whose coordinates are variable (so that the proof applies to all
shapes of that kind) but that is placed at a location in the coordinate plane that makes the
coordinate calculations relatively easy. For example, when proving theorems about triangles,
it is
often easiest to put one vertex at the origin and one side along an axis. You’ll see such a setup
used in the first problem. For other types of shapes as well, choosing a convenient location
makes proofs easier. For today’s assignment, each problem w
ill specify where the shape is
located. Eventually, you will be expected to make these decisions on your own.
Name:
Math 2
Date:
Problems
Directions:
Complete on separate paper. You may wish to use graph paper but it is not required.
1.
Consider
XYZ
with vertices at (0, 0), (
a
, 0),
and (
b
,
c
) respectively. Let
L
,
M
, and
N
be the
midpoints of the sides, as shown.
a.
Calculate the coordinates of
L
,
M
, and
N
.
b.
A segment between midpoints of two sides
of a triangle, such as
ML
, is called a
midsegme
nt
. Calculate the lengths of
midsegment
ML
and side
XY
.
How
do they compare?
c.
Calculate the slopes of midsegment
ML
and
side
XY
. How
do they compare?
d.
State the theorem(s) about triangle
midsegments that you have proved in parts
b
and
c
.
e.
Now state a
nd prove theorem(s) about
all
the sides of
LMN
in relation to
all
the sides of
XYZ
.
2.
For proofs about isosceles triangles (triangles with two equal sides), it’s easiest to put the
triangle’s line of symmetry along an axis.
a.
Draw a graph of a triangle with vertices at (
a
, 0), (
–
a
, 0), and (0
,
b
).
b.
Calculate distances to verify that this triangle is an isosceles triangle.
c.
Find the coordinates of the midpoints of the two equal sides.
d.
In triangles, a segment from a vertex to the midpoint of the opposite side is called a
median
. Prove tha
t in an isosceles triangle, two of the medians have equal lengths.
Name:
Math 2
Date:
3.
Consider the quadrilateral shown
in the diagram.
a.
To make this quadrilateral be
a parallelogram, what must
coordinate
d
equal (in terms
of
the other variables
a
,
b
,
and/or
c
)?
Hint:
In parallelograms,
opposite sides always have
equal lengths. Make lengths
WX
and
ZY
be equal.
For the rest of this problem,
assume that
d
is as you stated in
part
a
, making the quadrilateral
a
parallelogram.
b.
Calculate the coordinates of
the midpoint o
f diagonal
WY
.
c.
Calculate the coordinates of the midpoint of diagonal
ZX
.
d.
State the theorem you have just proved about the diagonals of a parallelogram.
4.
Consider the quadrilateral shown
in the diagram. Note that although
the coordinate labels ar
e the same
as before, we are no longer
assuming the fact about
d
that was
in problem
24a
.
a.
Explain why this diagram is
an appropriate setup for
proving theorems about
trapezoids.
b.
Let
M
be the midpoint of
ZW
and
N
be the midpoint of
YX
.
Calculate the s
lope of
MN
and
the distance
MN
.
c.
How does distance
MN
relate
to distances
ZY
and
WX
?
d.
State the theorem(s) you have just proved about trapezoids.
Name:
Math 2
Date:
5.
Here is an appropriate coordinate
setup to represent quadrilaterals in
general. It
does not assume t
hat the
quadrilateral has any special
properties.
Draw the midpoints of the four sides
and connect them to form
the
midpoint quadrilateral
.
Write a coordinate proof of this
theorem: “Given any quadrilateral,
the midpoint quadrilateral must be a
paralle
logram.”
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