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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