Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Lesson Title: The Development and Application of the
Distance Formula in Two Dimensions, with an Extension to Three Dimensions
Lesson Summary: Students will use inquiry-based learning to explore the mathematics involved in the
development and application of the distance formula. In part one of this lesson, students will begin to discover
a method for finding the lengths of horizontal and vertical segments. In part two, the students are asked to
answer a series of questions in order to investigate the lengths of oblique segments by making a connection to
the Pythagorean Theorem. This leads the students to a better understanding of the distance formula in twodimensions. In part three, the students will apply the distance formula, which they discovered in part two, in
order to develop the equation of a circle. In part three, the students are asked to extend their understanding of
the distance formula to three dimensions.
Key words: horizontal distance, vertical distance, oblique line, absolute value, right triangle, hypotenuse,
legs, Pythagorean Theorem, distance formula in both two- and three-dimensions, radius, circles, diagonal of a
prism, diagonal of a face
Background Knowledge:
We expect that the students have previous knowledge regarding: Pythagorean Theorem, Absolute Value,
plotting and reading points in the coordinate plane and basic algebraic simplification skills.
NCTM Standards Addressed:



Geometry Standard:
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop
mathematical arguments about geometric shapes
Specify locations and describe spatial relationships using coordinate geometry and other representational
systems
Use visualization, spatial reasoning, and geometric modeling to solve problems
Ohio Academic Content Standards:

Geometry and Spatial Sense
 Grade 8-10: G. Prove or disprove conjectures and solve problems involving two- and threedimensional objects represented within a coordinate system.

Mathematical Processes
 Formulate a problem or mathematical model in response to a specific need or situation, determine
information required to solve the problem, choose the method for obtaining this information, and
set limits for an acceptable solution.

Apply mathematical knowledge and skills routinely in other content areas and practical situations
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Learning Objectives:
1. SWBAT discover the distance between two points. This distance may be along a horizontal, a
vertical or an oblique path.
2. SWBAT develop the formula for finding the distance between any two points.
3. SWBAT apply the distance formula to discover the formula for the equation of a circle.
4. SWBAT extend the two-dimensional distance formula to three-dimensions.
Materials:
1. Pencil and Paper
2. Student Lesson Sheets
3. TI-Nspire or other graphing calculator or other graphing software
Suggested Procedure:
The following procedures may be modified to best fit your situation. We would allow the students to work in
groups of three or four students per group for one to three entire class periods.
1. Introduction: Find Distances in the classroom
Have three students stand in a right triangle in the room, following a horizontal line and vertical
line on the floor tiles. Ask the students to count the tiles they are apart. Why is it so difficult to
count the tiles between some of the students, while it is easier to count the tiles between other
students?
2. Part One of the Student Handout
Students will explore the length of vertical and horizontal lines. They should discover that in
order to find the distance you need to subtract either the y- or x- coordinates and then take the
absolute values.
3. Part Two of the Student Handout
Students will discover how to find the distance between any two points in two-dimensions. This
discovery, along with their previous knowledge of the Pythagorean Theorem, will lead them into
the development of the distance formula.
4. Part Three of the Student Handout
The students will apply the distance formula, which they discovered in Part Two, to a problem
situation in order to develop the equation of a circle.
5. Closure: Re-examine the Student Right Triangles
The three students will return to their locations to form the same right triangle in the room, which
began this lesson. Now ask the students to figure the distance between them.
6. Extension: Part Four of the Student Handout
In this extension, the students will extend the distance formula from two-dimensions into threedimensions.
Project AMP
Dr. Antonio R. Quesada
Assessments:
1. Student Lesson Sheets
2. Questioning techniques and debriefing during the inquiry activity
3. Group Work Participation
4. The Closure Activity
5. Quiz – Form A or Form B
Director, Project AMP
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Lesson:
Development and Application of the Distance Formula in Two Dimensions,
with an Extension to Three Dimensions
This lesson is designed to help the student develop and apply the distance formula in such a way that the
formula itself is intrinsically understood and not merely memorized.
Part One: Vertical and Horizontal Line Segments and Distance
Problems #1-3 refer to the following diagram.
1. How many units in length is the given line segment?
2. Label the upper endpoint of the line segment, A, and the lower endpoint, B.
a. What are the coordinates of point A?
b. What are the coordinates of point B?
3. Use the coordinates of points A and B to determine the length of the line segment.
Show your work.
4. Suppose the endpoints of a vertical line segment are C(-2, 1) and D(-2, 8). Use the coordinates of points
C and D to determine the length of the line segment. Show your work.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
5. Let points P(x1, y1) and Q(x2, y2) represent the endpoints of a vertical line segment.
a. What do we know about the values of x1 and x2?
b. How can you express the length, ℓ, of the line segment from P to Q?
c. Is there another way to express the length, ℓ, of the line segment from P to Q?
d. Will these answers in parts b and c necessarily be the same?
How are they the same? How are they different?
e. Write one expression that combines your answers from parts b and c.
Remember our discussions regarding absolute value.
f. Clearly explain why the absolute value is necessary in your expression for part e.
g. Would ℓ(P,Q) = | y1 - y2 | and ℓ(P,Q) = | y2 - y1 | be equivalent expressions? Explain.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
6. Suppose two points, P(x1, y1) and Q(x2, y2), lie on the same vertical line. We think of the distance, d,
between P and Q as the length of the line segment connecting P and Q.
a. Complete the equation below using one of your length answers from Question 5.
d(P,Q) =
b. Write this equation in a different manner using the other length answer from Question 5.
d(P,Q) =
c. Use either of these formulas to calculate the distance between points A and B from questions #1-3
above. Did you get the answer you expected? Explain.
7. Calculate the distance between the points P(5, -3) and Q(5, 7).
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Problems #8-10 refer to the following diagram.
8. How many units in length is the given line segment?
9. Label the left-hand endpoint of the line segment, A, and the right-hand endpoint, B.
What are the coordinates of point A?
What are the coordinates of point B?
10. Use the coordinates of points A and B to determine the length of the line segment.
Show your work.
11. Suppose the endpoints of a horizontal line segment are C(-0.8, 1) and D(-4.5, 1). Use the
coordinates of points C and D to determine the length of the line segment. Show your work.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
12. Let points P(x1, y1) and Q(x2, y2) be the endpoints of a horizontal line segment.
a. What do we know about the values of y1 and y2?
b. Express the length, ℓ, of the line segment from P to Q in such a way that the answer will be
appropriate whether x1 < x2 or x1 > x2. (You might want to refer to question #5.)
c. Write an expression that is equivalent to your answer in part b for the length, ℓ, of the line segment
from P to Q.
13. Suppose two points, P(x1, y1) and Q(x2, y2), lie on the same horizontal line. We think of the
distance, d, between P and Q as the length of the line segment connecting P and Q.
a. Complete the equation below using one of your length answers from Question 12
d(P,Q) =
b. . Write this equation in a different manner using the other length answer from Question 12.
d(P,Q) =
c. Use either of these formulas to calculate the distance between points A and B from questions #8-10
above. Did you get the answer you expected? Explain.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
14. Calculate the distance between the points P(-5.9, 11.4) and Q(2, 11.4).
15. Write out the two formulas for distance that have been developed in Part One of this lesson. Be sure to
differentiate between the horizontal and the vertical cases.
16. In Part Two of this lesson, you will need to square each of the expressions for distance that you have
written in Question 15.
a. Let us consider how that will look in the case of points P(x1, y1) and
Q(x2, y2) which lie on the same vertical line:
Square of the distance from P to Q: (| y2 - y1 |)2
True or False An equivalent expression for the square of the distance from P to Q is: (y2 - y1)2.
If True, justify your answer. If False, provide a counterexample. (That is, if False, provide an example
of specific values of y1 and y2 such that the two expressions produce different answers.)
b. Suppose two points R(x1, y1) and S(x2, y2) lie on the same horizontal line. Considering your work in
part (a), write the more simplified expression for the square of the distance from R to S.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Part Two: Finding Distances in the Coordinate Plane with the Distance Formula
1. Suppose you want to walk from point A (1,2) to point B (4,6). Using the sidewalks in order to stay off of the
grass, you start at point A and you decide to walk due east to point C (4,2). From point C, you then turn due
north and continue to walk to point B. Be careful of the graph scale.
a. At point C (4,2) where the horizontal and vertical paths meet, what is the angle measure at point C?
b. How far did you walk from point A to point C? (Let one full unit on the graph be equivalent to one step.)
c. How far did you walk from point C to point B?
d. How far did you walk all together?
e. You decide to take a different horizontal and vertical route from point A to point B.
Describe the direction you could walk.
f. What point did you use as the new point where your vertical and horizontal paths met?
g. What are your new horizontal and vertical distances? What is your new total distance?
h. Were these distances the same as the other distances you found earlier? Explain why or why not.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
2. Suppose now you decide to take a shortcut and walk directly from point A to point B, in a diagonal path.
Recalling the answers from the problem above, use your horizontal distance and vertical distance, along with
the angle measure at point C, to answer the following questions.
a. In the diagram above, draw lengths AC and BC. What specific type of polygon is ABC?
b. In this polygon, what specific name do we give to the side from point A to point B?
c. What theorem have we already learned that you can apply to this situation? State this theorem.
d. Using this theorem and the distances from the problem above, how far did you walk on the diagonal shortcut
from point A to point B? Is this really a shortcut? Explain.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
3. Examine the order pairs A (1,2) and B (4,6). Using your answers for the distances from the questions above,
which were figured by counting your steps walking due east and due north, is there another way to calculate
these distances in order to get the same answers?
In other words, how could you figure out the horizontal and vertical distances from point A(1,2) to point C(4,2)
and then to point B(4,6) without actually counting steps or looking at the graph? Explain your answers below.
a. Show the mathematical calculations you could use to figure out the horizontal distance from point A
to point C.
b. Show the mathematical calculations you could use to figure out the vertical distance from point C
to point B.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
4. Without graphing these two new given points, S (2, 4) and T (14, 20), and without counting the number of
steps, use the technique you discovered in the above problem to answer the following questions.
a. What is the horizontal distance across the x-axis to move from point S in the direction of point T?
b. What is the vertical distance up the y-axis to move from point S in the direction of point T?
c. What is the diagonal distance directly from point S to point T?
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
5. Suppose you want to start at any point D (x1, y1) and find the distance to any other point E (x2, y2). (Ignore
the scale of the graph.)
a. What expression could you use to find the horizontal distance from point D to a point F, that will become the
vertex of the right triangle? Keep in mind that distance must be positive. You might want to refer to your work
in Part One.
b. What expression could you use to find the vertical distance from point F to point E?
c. Using the general horizontal and vertical distances from point D to point F and then from point F to point E
from above, how could you find the “shortcut” or diagonal distance, d, from any point D to any point E?
Develop a single formula for calculating d. Make sure your formula is as simple as possible. Refer to Question
16 in Part One. The result will be “The Distance Formula”
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
6. Write the three expressions – one for the vertical case, one for the horizontal case, and one for the oblique
case – for finding distances that you have developed in Parts One and Two.
7. Could we use the distance formula that you developed for oblique lines to find the distance between two
points that lie on the same vertical line or on the same horizontal line? Explain or demonstrate.
8. For each problem below, calculate the distance between the two points.
State the exact answer and an approximate answer rounded to the nearest tenth. Show your work.
a. (8, -2) and (5, 4)
b. (3, 9) and (3, 1)
c. (-6, 2) and (2, -6)
d. (-10.3, -8.2) and (-12.9, -4.6)
e. (1.7, 0) and (5.7, -3.1)
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Part Three: Determining the Equation of a Circle
Start with a segment on the coordinate plane. Place one endpoint at the origin.
Let’s “lock down” the point located at the origin, but allow the other point to move freely, but at the same given
distance from the “locked down” point. Describe what you think will happen.
As a self-check, sketch by hand or use your calculator to create a sketch that either supports your conjecture or
provides a counterexample.
What have the outer endpoints of these segments created?
What name can we apply to the segments?
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Think about the work you have been doing with the Pythagorean Theorem and the Distance Formula. Write a
brief statement describing how you might find the length of the radius of this circle.
Let’s consider what has happened here so far.
 We have taken a segment and “locked down” one of its endpoints. That endpoint has become the center
of our circle. We are going to name the center with its own ordered pair called (h, k) because this point
will not change for a particular circle.
 The point that moves freely at a fixed distance from the center (h, k) around the circle will still be called
(x, y) because that point will change its x- and y-values as it moves.
 The distance between any point on the circle and the center of the circle has now become the radius of
our circle.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
The definition of a circle is the set of all points P(x, y) whose distance from the center C(h, k) is r (the radius).
So, point P is on the circle if and only if d(P,C) = r. (Notice that the center of the circle is not a point on the
circle. It is merely a point of reference.)
1. Using the distance formula, find the equation of the circle with center C(0,0) , point P(x, y) on the circle,
and radius of 5.
A more common form of the equation of a circle is found by squaring both sides of the equation you
have written above. Write the common form of the equation of the above circle.
Is the point (0, -5) a point on the circle? Verify your answer by showing that the point either does or
does not satisfy the equation of the circle.
Find another point on the circle. Verify your answer by showing that the point satisfies the equation of
the circle.
Graph this circle on your calculator. Explain why you needed to enter two equations to produce the
circle.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
2. Find the common equation of a circle with center C (2, 3), point P(x, y) on the circle, and radius of 5.
(Note: You do not need to expand the equation.)
3. Using the distance formula, develop the equation of a circle with center C(h, k) , point P(x, y) on the
circle, and radius of r. (Note: You do not need to expand the equation.)
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Extension: More Gems from the Math Treasure Chest
Three-dimensional distances
So far, you have reviewed the Pythagorean Theorem. You have used the legs and the hypotenuse of a right
triangle to help you make connections that have helped you discover how to find the distance between two
points on the coordinate plane. In other words, you have developed the Distance Formula.
Now let’s move a step forward.
The juice box pictured is 12 cm tall. The panel with the picture is 8 cm wide and the
side panel is 4 cm deep. Keeping the juice box upright as shown, what is the length of the
shortest straw that would fit inside the box? What is the length of the longest straw that
would fit inside the box? (Do not count the part of the straw that extends outside of the juice
box, only the part that would fit completely inside.) What do we call this kind of segment?
Draw a representation of this juice box on the graph paper below. Using your previous knowledge about the
Pythagorean Theorem and your development of the Distance Formula, determine a formula that allows you to
find the length of the longest straw that would fit inside the box. If you are on the right track, you should notice
something similar to your previous results. Verify this answer with your instructor before moving on to the
next question
Can you make a conjecture about the distance between two points in three-dimensional space?
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Test your conjecture by finding the distance between the following pairs of points. In each case, state the
equation you used in the first step. If you use the list menu on your calculator to create your results, state the
equation you used to generate the list of answers.
Point One
( 1, 1, 1)
(2, 3, 4)
(7, -2, 11)
(-6, -10, 5)
Point Two
(8, 8, 8)
(-4, -3, -2)
(0, 6, -12)
(-15, 20, -9)
Equation used
Distance between points
Extension Problem:
Flyfishing in
Wyoming
1. Gene and his son John are going fly-fishing in Wyoming and are buying an equipment locker for their
pickup truck so they can store their fishing rods in it. If the dimensions of one model of equipment locker are
4.5 feet by 1.5 feet by 1.75 feet, what is the longest fishing rod that will fit? Support your answer with a short
explanation stating how they can fit the rods into the locker and by showing your work.
2. The flatbed of their pickup truck can accommodate an equipment locker that fits snugly against the cab.
Because of this, any equipment locker they buy has to have a width of four and a half feet, but the depth and the
height can vary. Gene and John would like to buy new fly fishing rods. One model of rod is five and a half feet
long and another is six feet long. Fill in the missing dimensions with measurements to the nearest half inch for
the equipment locker in the tables below. If you use the list menu on your calculator to create your results, state
the equation you used to generate the list of answers.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Length of rod
5.5 feet
5.5 feet
5.5 feet
5.5 feet
Width of locker
4.5 feet
4.5 feet
4.5 feet
4.5 feet
Depth of locker
2 feet
2 feet 3 inches
2 feet 6 inches
2 feet 9 inches
Height of locker
Length of rod
6 feet
6 feet
6 feet
6 feet
Width of locker
4.5 feet
4.5 feet
4.5 feet
4.5 feet
Depth of locker
Height of locker
1 foot 4 inches
1 foot 7.5 inches
1 foot 10.5 inches
26 inches
3. The fishing rod that Gene and John would really like to take on their fishing trip is the FlyMaster FlexPro
that is six and a half feet long. Given the fact that the equipment locker cannot be longer than four and a half
feet in width and that they want to store the rods in the locker when they are on the road, they are trying to
determine different size lockers that might work so they can decide whether or not they can consider buying the
longer fishing rod. Help them out by finding some possible combinations of depth and height that will
accommodate the FlyMaster FlexPro.
Length of rod
6.5 feet
6.5 feet
6.5 feet
6.5 feet
Width of locker
4.5 feet
4.5 feet
4.5 feet
4.5 feet
Depth of locker
Height of locker
Project AMP
Dr. Antonio R. Quesada
NAME_______________________________________QUIZ FORM A__________
1. Find the distance between the points (2, 3) and (-8, 3). Justify your solution.
2. Find the distance between the points (8, 5) and (8, -6). Justify your solution.
3. Find the distance between the points (-2, 8) and (4, 6). Justify your solution.
4. Give the equation of a circle with center (4, 3) and radius of 3.
Director, Project AMP
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
NAME_______________________________________QUIZ FORM B__________
1. Find the distance between the points (2, 3) and (-8, 3). Justify your solution.
2. Find the distance between the points (8, 5) and (8, -6). Justify your solution.
3. Find the distance between the points (-2, 8) and (4, 6). Justify your solution.
4. Give the equation of a circle with center (4, 3) and radius of 3.
5. Challenge Problem: Lenny owns a lawn care business. In the back of his pickup truck, he has an equipment
locker that is 51 inches wide by 24 inches high by 18 inches deep. He has a couple of garden tools that are four
and a half feet long. A) Will they fit into his equipment locker? B) At most, what is the longest garden tool
that might fit into his locker?
6. Bonus: Consider the ideas you have developed about two- and three-dimensional distances. Write an
equation that you think could be used to generate a sphere.