Honors Math 2
Name:
Date:
Coordinate Proofs
Objective: Prove geometric theorems using coordinate methods.
Introduction to coordinate proofs
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 the same point.
These occurrences were often an indication that there is a geometric theorem applied to that type
of diagram. We are now turning our attention to identifying these theorems and writing proofs of
them, using the machinery of coordinate calculations. In some cases, these are theorems we have
already proven deductively.
The idea of a coordinate proof 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 (x 2  x1)2  (y 2  y1)2 and comparing them.

Facts about parallel lines or perpendicular lines can be proved by calculating slopes using
y 2  y1
and comparing them (equal slopes indicate parallel; slopes having a product of –1
x 2  x1

indicate perpendicular).

Facts concerning midpoints can be proved by using the midpoint coordinates
x1  x 2 y1  y 2 
,

in appropriate distance or slope calculations.
 2
2 

Facts concerning points forming other ratios can be proved using the formulas we devised at our
last class. For example, the point that’s 23 of the way from (x1, y1) to (x2, y2) can be calculated as
( 13 x1 + 23 x2, 13 y1 + 23 y2). For other fractions, replace 23 and 13 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
work with a shape whose coordinates
 is to 
 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 will specify where the shape is
located. Eventually, you will be expected to make these decisions on your own.
Example
Theorem: The midpoint of the hypotenuse in a right triangle is equidistant from the three vertices
of the triangle.
Proof:
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
midsegment. 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 and prove theorem(s) about all
the sides of LMN in relation to all the sides of XYZ.


Z(b,c)
M
X(0,0)
L
N
Y(a,0)
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 that in an isosceles triangle, two of the medians have equal lengths.
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.
Z(b,c)
W(0,0)
Y(d,c)
X(a,0)
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 of 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 are the same
as before, we are no longer
assuming the fact about d that was
in problem 3a.
a. Explain why this diagram is
an appropriate setup for
proving theorems about
trapezoids.
Z(b,c)
W(0,0)
Y(d,c)
X(a,0)
b. Let M be the midpoint of ZW
and N be the midpoint of YX.
Calculate the slope 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.
5. Here is an appropriate coordinate
setup to represent quadrilaterals in
general. It does not assume that the
quadrilateral has any special
properties.
Z(b,c)
Y(d,e)
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
parallelogram.”
W(0,0)
X(a,0)