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PYTHAGOREAN THEOREM – MORE OR LESS
Teacher Edition
List of Activities for this Unit:
ACTIVITY
STRAND
DESCRIPTION
ME
Walkthrough of the Pythagorean
Thm
Applying the Pythagorean Thrm
Exploring when values a² + b² is
greater or less than c²
Applying the Pythagorean Thrm
Similar figures
Similar figures
Giving support for why figures are
similar
Apply concepts to building
situations
Review and practice
1 - Pythagorean Theorem
2 – The Tennis Court
ME
ME
3 – It’s all squares to me
4 – Fire, Fire
5 – Are you similar to me?
6 – Similar Things
ME
GS
GS
GS
7 – Appearance is Everything
ME/GS
8 - Common Truss
ME/GS
9 - Pythagorean Thrm. MC practice
COE Connections
MATERIALS
Grid/Graph Paper
Protractor
Colored card stock
Geoboard(s)
Rulers
Vocabulary: Mathematics & ELL
acute
altitude
appear
base
congruent
coordinates
Corresponding
diagonal
diagrams
dimensions
equal
Teacher: Ch 14 “Pythagorean Theorem”
equivalent
figure(s)
framework
horizontal
hypotenuse
intersection
legs
obtuse
ordered pair
parallel
perpendicular
pitch
proportionally
random
relationship
scale drawing
Scale Factor
similar
squares
surveillance
vertex
vertical
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Essential Questions:
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What is a right triangle?
What are the legs and hypotenuse of a right triangle?
What does the word bisect mean?
What is the Pythagorean Theorem?
How can the Pythagorean Theorem be used to find the missing lengths of the sides of a right
triangle?
How can the Pythagorean Theorem be used to determine if a given triangle is an acute, an obtuse,
or a right triangle?
What does similarity mean?
Do we know that two figures are similar just because they appear to be similar?
How do you determine when two triangles are similar?
How are ratio and proportion used to solve problems that deal with similarity?
How do you determine the perimeter and area of a triangle?
What is meant by the scale of a diagram?
What is the altitude of a triangle?
What are the properties of an altitude of a triangle?
What are the properties of an altitude of isosceles triangles?
Lesson Overview:

Before allowing the students the opportunity to start the activity: access their prior knowledge
with regards to properties of triangles (altitudes, area, perimeter, etc.), parallel lines and scale
factor. Warm-up exercises, discussion in collaborative groups, problems on the wall around the
room can be used.
 Discuss altitudes of triangles before starting question #11.
 What is being asked by the questions in the problem?
 How can the students make their thinking visible?
 Use resources from your building. Technology could be used to support this investigation—
especially sketchpad or cabri geometry.
The Common Trusses can be used to show the students various examples of trusses.
Performance Expectations:
7.2.C Describe proportional relationships in similar figures and solve problems involving similar
figures.
7.2.D Make scale drawings and solve problems related to scale.
7.2.E Represent proportional relationships using graphs, tables, and equations, and make connections
among the representations.
8.2.F Demonstrate the Pythagorean Theorem and its converse and apply them to solve problems.
8.2.G Apply the Pythagorean Theorem to determine the distance between two points on the coordinate
plane.
8.4.C Evaluate numerical expressions involving non-negative integer exponents using the laws of
exponents and the order of operations.
G.3.D Know, prove, and apply the Pythagorean Theorem and its converse.
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Performance Expectations and Aligned Problems
Chapter 14
“Pythagorean
Theorem”
Subsections:
Problems
Supporting:
PE 7.2.C ≈
7.2.E
Problems
Supporting:
PE 7.2.D
Problems
Supporting:
PE 8.2.F ≈
8.2.G ≈
G.3.D
Problems
Supporting:
PE 8.4.C
1-
2-
3-
4-
5-
6-
7-
8-
9-
10-
Pythagorean
Theorem
The
tennis
Court
It’s all
Squares
to me
Fire, Fire
Are You
Similar
to me?
Similar
Things
Appearance is
Everything
Common
Truss
Geoboard
Triangles
MC
Problems
3-5
1, 2
3–9
10
11
12, 13
3–9
10
11
12, 13
2-4
1-4
1, 2
1, 2
11
15
3, 4
1–4
1, 2
1, 2
11
15
Assessment: Use the multiple choice and short answer items from Measurement and Geometric Sense
that are included in the CD. They can be used as formative and/or summative assessments attached to this
lesson or later when the students are being given an overall summative assessment.
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1 – Pythagorean Theorem
1. What is a right triangle? A triangle with one right angle; a 90º angle.
_________________________________________________________________________________
2. Right triangles have special names for their sides. The two sides that come together forming the right
angle are called legs. The side opposite the right angle is called the hypotenuse. The hypotenuse will
ALWAYS be the longest side. (Why?) The side opposite the smallest angle will ALWAYS be the
shortest leg. (Why?)
3. The table gives the lengths of the legs and hypotenuse of various right triangles. Complete the table.
Short
Leg
Long
Leg
Hypotenuse
(Short
Leg)²
(Long
Leg)²
(S Leg)² + (L Leg)²
Hypotenuse²
3
4
5
9
16
25
25
9
12
15
81
144
225
225
6
8
10
36
64
100
100
8
15
17
64
225
289
289
a. What do you notice in the table?
The last two columns are the same.
The sum of the two legs squared is equal to the hypotenuse squared
b. This relationship only exists in right triangles and is called the Pythagorean Theorem.
Write in your own words a description of the Pythagorean Theorem. ____________________
The sum of the two legs squared is the hypotenuse squared.
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4. Remember that (Short Leg)² + (Long Leg)² = Hypotenuse² and by algebra we also derive
(Short Leg)2 = Hypotenuse2 – (Long Leg)2 OR (Long Leg)2 = Hypotenuse2 – (Short Leg)2
a.
Determine the length of the missing side in each triangle; round your answers to the nearest
tenth of a unit.
Show your work using words, numbers and/or diagrams (charts or graphs).
c = ______41 in______
c = ____29.6816 mm_______
= 41 in
Same kind of work as above, different numbers
b = ___17.3494 ft_______
Teacher: Ch 14 “Pythagorean Theorem”
= 17.3494 ft
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2 - The Tennis Court
1. A tennis court measures 36 feet by 78 feet. What is the diagonal’s (a diagonal is the straight line
from one vertex of a figure to an opposite vertex.) length?
Show your work using words, numbers and/or diagrams.
= 85.9069 ft
2. A major league baseball diamond measures 90 feet by 90 feet. Catchers are often required to throw
from home plate to 2nd base. How far is the throw from home plate to 2nd base?
Show your work using words, numbers and/or diagrams.
= 90
Teacher: Ch 14 “Pythagorean Theorem”
ft = 127.2792 ft
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3. An eight-meter ladder leans against a building. The base (bottom) of the ladder is three meters from
the base of the building. How far is it from the top of the ladder to the base of the building?
Show your work using words, numbers and/or diagrams.
? =
=
= 7.4161 m
4. A 20 m pole will be held by a wire extending from the top of the pole to a stake driven into the ground
15 m away from the base of the pole. What is the length of the extended wire?
Show your work using words, numbers and/or diagrams.
= 25 m
NOTE: It is assumed that the intersection of the pole
and the ground is a right angle.
20 m
Wire
The students most likely will
have to draw a picture to get
this problem solved.
15 m
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3 – It’s All Square to Me
1. Using grid paper, cut out squares with areas of 9, 16, 25, 36, 81, 100, 121, 144, 169 and several more
of your choosing. Use three of the squares to construct a triangle as shown below. Tape the squares in
position on the colored paper. Create 5 different such triangle sets as shown, but different sizes.
a. Determine if the triangle formed is acute (measures between 0º and 90º) , right (measures exactly
90º) or obtuse (measures between 90º and 180º) by measuring the angles. Record your data
(putting the lengths of the sides of the triangle in increasing order) in the accompanying chart.
Continue this process until you have measured and recorded 5 sets of squares and entered the data
into your chart. In your data find a pattern similar to the Pythagorean Theorem which describes
the relationship between the squares of the lengths of the sides of a triangle and the type of triangle
formed.
Triangle
Area of the Area of one Area of the
side lengths
largest
of the
remaining
in
square
smaller
square
increasing
square
order
a, b, c
c2
a2
b2
5, 6, 10
100
Is the
triangle
formed
acute, right
or obtuse?
a 2 + b2
25
36
61
Obtuse
Example Answers
4, 5, 6
36
16
25
41
Acute
3, 4, 5
25
9
16
25
Right
b. For what type of triangle is c² = a² + b²? _____A right triangle_____________
c. If c² ≠ a² + b², for what kind of triangle is c² < a² + b²? ___Acute________
d. If c² ≠ a² + b², for what kind of triangle is c² > a² + b²? __Obtuse____
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2. In your own words, state the “Un-Pythagorean Theorem”. ________________________________
If c² is greater than a2 + b², then the triangle is obtuse.
If c² is less than a² + b², then the triangle is acute.
Right triangle ABC has a right angle at vertex C. Leg a = 30 inches and Leg b = 40 inches.
a. What is the length of side c, the hypotenuse? _____________50 in_________
Show your work using words, numbers and/or diagrams.
= 50 in
b. If triangle ABC from above was an acute triangle, what would you know about the length
of side c?
c would be less than 50 inches.
c. What if angle C were an obtuse angle what would you know about the length of side c?
c would be more than 50 inches.
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4 – Fire, Fire
1.A surveillance (observation) helicopter reported a grass fire to the central dispatch. The dispatcher
located the fire on a gridded map and determined the fire is 8 miles due east and 11 miles due north
from the central dispatch (radio room). The dispatcher (radio operator) establishes the fire at
coordinates (8, 11). The dispatcher determines that the fire station located at coordinates (2, 3) is
closest to the fire and calls that station to send out a truck. The fire truck travels in a direct route with
no obstacles.
Fire Station
Central Dispatch
Fire
a. What are the coordinates (ordered pairs of numbers that identify points on a plane) of the
right angle vertex of the triangle? (8, 3)
b. What is the horizontal (extending from side to side; parallel to the horizon) distance between
the station and the fire? 6 miles
Show your work using words, numbers and/or diagrams.
8 miles – 2 miles = 6 miles
c. What is the vertical (extending straight up and down; perpendicular to the horizon) distance
between the fire station and the fire? 8 miles
Show your work using words, numbers and/or diagrams.
11 miles – 3 miles = 8 miles
d. What is the actual distance from the station to the fire?
10 miles
Show your work using words, numbers and/or diagrams.
= 10 miles
e. A fire truck in that area travels at an average speed of 50 mph? How long will it take, in
minutes, for a fire truck to get from the fire station to the fire? 12 minutes
Show your work using words, numbers and/or diagrams.
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d = r•t
•
=
= 12 min
2. A surveillance helicopter reported another grass fire to the central dispatch. The dispatcher located the
fire on a gridded map and determined the fire is directly west of the central dispatch. The dispatcher
determines that the fire station located at coordinates (-1, -7) is closest to the fire and calls that station
to send out a truck. The fire truck travels in a direct route with no obstacles (interferences).
a. What are the coordinates of the right angle vertex (corner) of the triangle? (-1, 0)
b. What is the horizontal distance between the fire station and the fire?
Show your work using words, numbers and/or diagrams.
=
= 6 (distance is never negative)
c. What is the vertical distance between the station and the fire?
Show your work using words, numbers and/or diagrams.
=
= 7 (distance is never negative)
6 miles
7 miles
d. What is the distance from the station to the fire?
9.2195 miles
Show your work using words, numbers and/or diagrams.
≈ 9.2195
e. A fire truck in that area travels at an average speed of 50 mph? How long will it take, in
minutes, for a fire truck to get from the fire station to the fire?
11.06 minutes
Show your work using words, numbers and/or diagrams.
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d = r•t
•
≈ 11.06
3. A scientist determined that twins occur in 1 out of every 80 births. The scientist uses a random
(chance) sample of 560 pregnant women.
Which represents the number of pregnant women expected to give birth to twins?
 A.
7
 B.
8
 C.
70
 D.
80
4. A commercial artist has a sketch of a rectangular logo that is 7 inches high. She needs to proportionally
(a part to a whole comparison) reproduce the logo on a sign that is 8 feet high. The sketch of the logo
contains a letter M that is 5 inches tall.
Which represents how tall the letter M will be on the larger sign?
 A.
4.4 feet
 B.
5.7 feet
 C.
6.0 feet
 D.
11.2 feet
5. Damon wants to fertilize his lawn for the spring season. The dimensions (size) of his lawn are shown.
Johnson’s Garden Shop sells fertilizer in 6-pound bags that cover an area of 500 square feet.
Which represents the number of bags of fertilizer Damon will need to completely fertilize his lawn?
 A.
2
 B.
3
 C.
4
 D.
5
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5 – You are Similar to Me
1. If someone says to you that you look similar to someone else, what do they mean? Probably they mean
that you look alike, almost the same. In geometry, similar figures look alike. They have the same
shape, but they may not be the same size. Have you ever used a copy machine? A copy machine can
enlarge or reduce the size of what is being copied. Here are some triangles that have been copied:
a. Which triangles are similar to triangle A?
All of them!
They all have the same shape, but not all have the same size.
___________________________________________________________________________
b. Does it matter that some of the triangles are turned sideways or upside down?
Orientation does not affect similarity.
c. Do they still have the same shape?
No
Yes
d. Figures that are exactly the same size and shape are called congruent. Are any of the triangles
shown congruent? Yes
If so list the congruent triangles: A, B, D
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2. Use these figures (shapes) to answer the following questions:
a. How are the figures alike?
They are all triangles: having three sides and three angles.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
b. How are the figures different? The lengths of the sides vary.
The corresponding angles for any given two triangles are not congruent.
___________________________________________________________________________
___________________________________________________________________________
 In order for figures to be similar these two conditions listed must be met:
Condition: Corresponding (or matching) angles must have the same measure.
AND
Condition: Corresponding sides must all be in the same ratio.
c. Are the figures all similar?
No
Show your work using words, numbers, and/or diagrams
None of the triangles are similar to each other because the angles are not congruent and
the sides are not proportional.
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6 – Similar ThingsType equation here.
3. Use these figures (shapes) to answer the following questions:
This is the symbol for
“corresponds to”.
a. Which are the matching or corresponding sides in the two rectangles?
,
b. Which are the matching or corresponding angles? A
E, B
F etc.
c. Are the rectangles similar? Yes, if we agree all the angles in both shapes are right angles.
(Reference (look at) both sides and angles)
4.Use these two triangles to answer the following questions:
a. If there are corresponding sides in the two triangles list them as pairs: ____________________
,
,
Looking at each corresponding pair, is there a common ratio? Yes, 4 times the smaller
triangles side is the length of the corresponding side on the larger triangle.
If so what is the common ratio?
ratio is .
Smaller to larger the ratio is 4. Larger to smaller the
b. If there are the corresponding angles List them as pairs: A
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E, C
D, B
F
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c. Are the triangles similar? YES Explain why or why not. Corresponding sides are in the
same ratio.
5. Use these shapes to answer the following questions:
If you know that two figures are similar, you can use proportions or equivalent (equal) ratios to find
the lengths of the unknown sides.
a. The two rectangles shown are similar. What would be the proportion that you would use to
determine the length of the missing side?
=
b. Solve the proportion for x. 6 cm
Show your work using words, numbers, and/or diagrams.
=
x =
= 6cm
6. Tell whether the pair of polygons in figure 1 are similar and whether the polygons in figure 2 are
similar.
a. Are the polygons in figure 1 are similar? NO Why? Corresponding sides do not have the same
ratio.
b. Are the polygons in figure 2 are similar? NO Why? Corresponding sides do not have the
same ratio.
Figure
Figure
Show your work using words, numbers, and/or diagrams.
Figure 1: 3(2cm) = 6cm, but 3(4cm) ≠ 8cm
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Figure 2: 3(5m) = 15m, but 3(3m) ≠ 10m
7. The right triangles shown are similar. Determine the length of the missing sides.
x = 4m y = 10m z = 8m
n = 20m p = 12m
Show your work using words, numbers, and/or diagrams.
Let the three triangles be called Small, medium, and large:
The ratio from small to medium is 2 and the ratio from small to large is 4.
By the Pythagorean Theorem, x = 4… so the small triangle has sides of 3,4,5  the medium
has sides of 2(3,4,5) = 6,8,10, and the large has sides of 4(3,4,5) = 12, 16, 20.
8. The water skiing ramps shown have similar triangles. What is the height of the smaller ramp?
6 ft
Show your work using words, numbers, and/or diagrams.
24ft (?) = 8ft  ? =
so 18ft • = 6ft
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9. A surveyor could use similar triangles to determine the distance (d) across the lake.
What is the distance across the lake? d = 9 miles
Show your work using words, numbers, and/or diagrams.
The two triangles are similar because both triangles have a right angle and the two lines that cross
form vertical angles (meaning they are equal in measure) and the remaining angles are congruent
because triangles have 180º. Similarity establishes that corresponding sides are in a fixed ratio.
Pythagorean Theorem on the small triangle yields 13 miles – 4 miles = 9 miles  the missing side
is 3 miles. The small triangle’s 2 mile side corresponds to the 6 mile side of the large triangle; this
indicates that the scale factor for similarity is 3. Therefore, the small triangle’s side of 3 miles
times 3 is the length “d” in the large triangle; ie 9 miles.
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7 – Appearance is Everything
10. Examine each of the shapes below. For each pair, decide if the two shapes appear to be similar
without using any measurements. Use a ruler and protractor to make measurements to help you
decide if the shapes are similar or not. Measure all lengths in centimeters units. Record your
measurements on the figures and show any calculations that were performed.
a. Do the these triangles appear (look) to be similar? Yes or No
B
A
Measurements/Calculations:
Triangle A has side lengths all measuring 1.8cm and triangle B has all sides measuring 4.6cm.
Both triangles have all angles measuring 60°.
Are they really similar? Yes or No? Support your conclusion. Yes the angles are congruent
(and the sides are proportional).
b. Do the these triangles appear to be similar? Yes or No
AB =4cm BC=2cm
AC=4.5cm
mA=26°
mB=90°
A
FD=2cm FE=2cm
DE=2.8cm
mC=64º
D
B
C
mD=45°
mF=90°
mE=45º
E
F
Measurements/Calculations:
Are they really similar? Yes or No? Support your conclusion. Angles are not congruent.
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c. Do the these triangles appear to be similar? Yes or No
7.7 cm
83°
.9 cm
2.5 cm
2.6 cm
83°
19°
19°
7.8 cm
79°
Teacher Note:
2.6 cm
Due to variations in
measurement, the ratios of the
corresponding sides may not be
exactly equal.
79°
Measurements/Calculations:
Great class discussion: Are the
ratios close enough to be
considered proportional?
Are they really similar? Yes or No? Support your conclusion. The angles are congruent.
≈
≈
(At least they are very close to the same ratios!)
d. Do these squares appear to be similar? Yes or No
2.1 cm
3.1 cm
Measurements/Calculations:
All squares have four 90° angles.
=
=
=
Are they really similar? Yes or No? Support your conclusion. All angles are congruent and
all sides have the same ratios.
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e. Do these rectangles appear to be similar? Yes or No
5.5 cm
1.8 cm
2.7 cm
3.6 cm
Measurements/Calculations:
All rectangles have four 90° angles.
≈
(At least they are very close to the same ratios!)
Are they really similar? Yes or No? Support your conclusion.
f. Write a description of what it means for two shapes to be similar.
In order for figures to be similar these two conditions listed must be met:
Condition: Corresponding (or matching) angles must have the same measure.
AND
Condition: Corresponding sides must all be in the same ratio.
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8 – Common Trusses
Teacher Note:
This can be used as an
introduction to problem 11.
What Mathematics/Geometry
is involved in the construction
of trusses?
A truss is a rigid framework (structure) used in building
bridges and roofs. Trusses are an efficient way to span long
distances with a minimum of materials and still maintain
strength. The following are three commonly used trusses.
Triple Howe Agricultural Truss: Depending on pitch (the ratio of vertical change to horizontal
change) and spacing, these trusses can clear span up to 84 feet.
2-piece “Piggyback” trusses can achieve steep pitches over large spans.
Common attic truss can provide “Bonus Room” over garage or elsewhere.
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11. The pitch of a roof is defined as the ratio of vertical change to horizontal change (also known as
slope). Below is a diagram of a roof with a 2/4 pitch spanning 24 feet (segment
).
Segment
= Segment
(diagram is not drawn to scale)
A
B
R
The altitude (height) from point S is a perpendicular segment drawn to side
triangles, the altitude bisects (cuts in half) both the angle and the side.
. In isosceles
a. Draw the altitude from point S and label this intersection (the point where two lines cross or
touch) point R.
b. Determine the length of segment
12 ft
Show your work using words, numbers, and/or diagrams.
• 24ft = 12ft
c. Determine the length of segment . 6 ft
Show your work using words, numbers, and/or diagrams.
∆ABP ∆SRP and the scale factor is three. Three times the smaller is the larger.
This means the height PR is: 3•2ft = 6 ft.
d. Determine the length of segment . Express your answer in terms of feet and inches.
Show your work using words, numbers, and/or diagrams.
(PS)2 = (12 ft)2 + (6 ft)2
(PS)2 = 180 ft2
PS = 6
ft ≈ 13.4164 ft
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e. Make a scale drawing (smaller/larger version of original drawing) of the (Triple Howe
Agricultural Truss) roof on the grid. Indicate the scale factor that you chose. Be sure to label
the vertices with the letters P, S and Q in the appropriate locations to match the roof. Also, be
sure to label R.
Scale Factor (ratio expressing the amount of magnification):
each square is 2ft x 2ft.
f. Subdivide
into four equal segments. Label each subdivision point from left to right A, B,
and C. At each subdivision point draw a perpendicular to
. Label the intersection of each
perpendicular (at right angles to the horizon) with PS from left to right as: L, M and N.
Determine the lengths of segments ,
and
.
Show your work using words, numbers, and/or diagrams.
∆PAL ∆PBM ∆PCN ∆PRS and mPA = 3ft  mAL = 1.5 ft
scale factor is 2 times ∆PAL = ∆PBM  mBM = 3 ft
= 2•1.5 ft
scale factor is 3 times ∆PAL = ∆PCN  mCN = 4.5 ft = 3•1.5 ft
scale factor is 4 times ∆PAL = ∆PRS
g. Draw segments ,
, and
By the Pythagorean Theorem:
mLB ≈ 3.3541 ft
mMC ≈ 4.2426 ft
mNR ≈ 5.4083 ft
 mRS = 6 ft = 4• 1.5 ft
. Determine these lengths.
h. To complete the truss, reflect the figure across
above.
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. Answer will look like the Howe Truss
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9 – Geoboard Triangles
12. Examine the three triangles on the geoboard: triangle CDB, triangle CEG and triangle CFA.
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a. Are the three triangles similar? YES
b.
Why or why not? All are right triangles with the common angle at C; this means all the angles
are congruent and by Pythagorean Theorem the corresponding sides are in Proportion.
c. Turn the geoboard so that the legs are vertical and horizontal. Find the ratio between the
vertical and horizontal legs of each right triangle. SAMPLE ANSWERS: Using the geoboard
as oriented above, the ratio is . If the line segment CF is oriented to vertical(C at the top),
then the ratio is .
What can you say about this ratio? The ratio contains the numbers for the slope but not the
sign on the numbers for the slope
Teacher: Ch 14 “Pythagorean Theorem”
28
6/18/08
Page 26 of
13. You investigated a nest of three right triangles that shared the common acute angle at C. Create a nest
(one triangle inside the other – similar to the example given) of similar triangles that are not right
triangles. How do you know that the three nested triangles are similar?
Sample answer given: angle A is the same for all the nested triangles, Lines GB, FC, & ED have the
same slope as ED. This means the angles at G, F, & E are all congruent. A further implication is the
angles B, C, & D are congruent, because all triangles have 180° and in each triangle two of the angles
are the same implies the remaining angles in the triangles must also be the same.
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Teacher: Ch 14 “Pythagorean Theorem”
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Page 27 of
10 - Practice Problems
14. Lolla thought that she remembered a relationship between the lengths of the sides of a triangle and the
measure of the angles across from the sides.
Which represents the relationship of the angles of PQR from smallest to largest?
 A.
R, Q, P
 B.
R, P, Q
 C.
Q, R, P
 D.
P, R, Q
15. The club members hiked 13 kilometers north and 14 kilometers east, but then went directly home as
shown by the dotted line.
Which is the distance they traveled to get home?
 A.
5.2 km
 B.
15.0 km
 C.
19.1 km
 D.
27.0 km
Teacher: Ch 14 “Pythagorean Theorem”
28
6/18/08
Page 28 of