# The Pythagorean Theorem and Special Right Triangles

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```					Project AMP                      Dr. Antonio R. Quesada          Director, Project AMP

Lesson Plan: Pythagorean Theorem and Special Right Triangles

Lesson Summary
This lesson will provide students with the opportunity to explore concepts including
the Pythagorean Theorem, rational and irrational numbers, and the relationship of
side lengths of special right triangles.

Key Words
 Hypotenuse
 Irrational Numbers
 Legs
 Pythagorean Theorem
 Rational Numbers
 Right Triangles

Background Knowledge
We are assuming the students are proficient at calculating square roots and squares,
solving multi-step equations as well as basic quadratic equations. We are also
assuming they have a strong background in basic geometry concepts including
finding area and perimeter and classifying angles. Lastly, we assume students have
basic graphing calculator proficiency in finding squares, finding square roots, and
using the table function.

 Geometry and Spatial Sense Standard: Students identify, classify, compare
and analyze characteristics, properties and relationships of one-, two- and
three-dimensional geometric figures and objects. Students use spatial
reasoning, properties of geometric objects and transformations to analyze
mathematical situations and solve problems.
o Grade 7: Characteristics and Properties
 Indicator #3: Use and demonstrate understanding of the
properties of triangles.
For example:
a. Use Pythagorean Theorem to solve problems involving right
triangles.
b. Use triangle angle sum relationships to solve problems.
o Grade 8: Characteristics and Properties
Project AMP                      Dr. Antonio R. Quesada           Director, Project AMP

  Indicator #1: Make and test conjectures about characteristics and
properties (e.g., sides, angles, symmetry) of two-dimensional figures
and three-dimensional objects.
   Number, Number Sense and Operations Standard: Students demonstrate
number sense, including an understanding of number systems and operations
and how they relate to one another. Students compute fluently and make
reasonable estimates using paper and pencil, technology-supported and mental
methods.
o Grade 7: Number and Number Systems
 Indicator #3: Describe differences between rational and irrational
numbers; e.g., use technology to show that some numbers
(rational) can be expressed as terminating or repeating decimals
and others (irrational) as non-terminating and non-repeating
decimals.

Learning Objectives
 Understand Pythagorean Theorem
 Apply Pythagorean Theorem to real-world scenario
 Understand the difference between rational versus irrational numbers
 Recognize the relationships of side lengths in special right triangles
 Apply knowledge of special right triangles to real-world scenarios

Materials
 Ziploc bags containing colored straws of different lengths. Use lengths so that
not all combinations will result in a triangle.
 Calculators
 Activity and extension activity sheets
 Pencils

Suggested Procedure
 The time frame for this lesson is 2 days.
 Day 1
o Group students into heterogeneous groups of 3.
o Attention Getter: Pass out Ziploc bags with straws to students and pose
the question “Can you get a right triangle from any 3 side lengths?” Ask
students to work individually on this question and then discuss their
findings with their group members. Facilitate class discussion after 2
minutes.
o Pass out Pythagorean Theorem activity sheet and have students work on
this activity sheet within their groups. Walk around and observe.
Project AMP                         Dr. Antonio R. Quesada            Director, Project AMP

Remember to let students determine conjectures and solutions – do not
give answers! After completion of this activity sheet, facilitate a debrief
discussion of the activity.
o     Ask students to summarize individually in their math journals what they
learned today.
   Day     2
o      Pass out Extension Sheets on 30°, 60°, right triangles and 45°, 45°, right
triangles.
o     Have students work in same groups on these activities.
o     At conclusion of activities, debrief as a whole class.
o     Ask students to summarize individually in their math journals what they
learned today.

Assessments
 Observation
 Class Discussion
 Written solutions and explanations
 Math Journals
 Future quiz and/or test
Project AMP                         Dr. Antonio R. Quesada              Director, Project AMP

The Pythagorean Theorem and Special Right Triangles

Group Members ____________________________                   ______________________________

____________________________             ______________________________

Before We Begin
Can a right triangle be formed using any three lengths of sides?
Select three straws from the bag. Use the corner of a piece of notebook paper for a
right angle. Use two of your straws placed along the edges of your notebook paper as the legs
of the right triangle. The third straw will form the hypotenuse of the right triangle. Do your
straws form a right triangle? Compare your results with those of your members. How many
of you were able to form a right triangle?
No, two group members in our group did not have straws that fit together to
form a right triangle. One of those two group members was able to form an acute
triangle. But the last group member was not able to form a triangle at all. So, our group
concluded that there are certain side lengths that form right triangles and others that
do not.

Lesson 1
We recall the area formula for a square is A = s2. Using the dimensions given, complete
the table by finding the area of the squares in the diagram below.

1.)

a          b          c           a2         b2           c2

3          4          5           9          16           25

7          10         149        49          100         149

6          8          10         36          64          100

5          171        14         25          171         196
Project AMP                                   Dr. Antonio R. Quesada               Director, Project AMP

2.) What relationship do you notice with the areas of the three squares?
The sum of the areas of the smaller two squares is equal to the area of the largest
square.

3.) Based on the patterns observed in the table, what conjecture can you make?
a2 + b2 = c2

4)    Notice two of the dimensions in the table are expressed as radicals, why do you think
they are expressed as radicals and not in decimal form?
The decimal form does not end. It keeps going.

5) When writing the decimal form of these numbers, is it possible to give an exact answer?
Why or why not?
No, it isn’t. Since the decimal doesn’t end, you have to round. Therefore, the answer
is approximate.

Definition:
Irrational number – An irrational number is a number that cannot be expressed as a
fraction    for any integers and . Irrational numbers have decimal expansions that neither
terminate nor repeat.
For the remainder of this lesson, you may want to express irrational answers as decimal
approximations (for real world applications) or in simplest radical form (to express a more
precise value).

a = 1, b = 2, c =        5 . Yes, since a2 = 1, b2 = 4, and c2 = 5 and 1 + 4 = 5 our conjecture
holds.

7.) Now, use that relationship to find the missing length in the right triangles below.

119 ft

60 ¾ in
57 ft

164.01 cm
b
100 cm                                                c

57 in
a
a. )                      b.)                                             c.)

a = 129.997                       c = 131.947                               b = 21.0134
Project AMP                            Dr. Antonio R. Quesada             Director, Project AMP

8) Summarize, in your own words, what you have learned today.

   The sum of the areas of the smaller two squares is equal to the area of the largest
square.
   a2 + b2 = c2 is the Pythagorean Theorem
   The decimal form of irrational numbers keeps going.
   The decimal form gives an approximate answer.

Real World Application: Play Ball!

2nd Base

90 ft.                  90 ft.

X

3rd Base                                                 1st Base

90 ft.                               90 ft.

Home
Plate

The distance from each consecutive base is 90 feet and it can be necessary to
determine how far the catcher will have to throw to get the ball from home plate to
2nd plate or the distance of the throw from the 3rd baseman to 1st base. We will use
the Pythagorean Theorem to find answers to these questions.
Project AMP                            Dr. Antonio R. Quesada                 Director, Project AMP

1. What other shape is the baseball diamond?

Square

2. Explain how you might use the Pythagorean Theorem to determine a value for x.

3. Where is a right triangle located?

The dotted line divides the baseball diamond into 2 right triangles.

4. Determine what side of the right triangle is the hypotenuse.

The hypotenuse is side x.

5. Determine the legs of the triangle.

The legs would be the side from home plate to 1st base and the side from 1st to 2nd base.
Or
The legs would be the side from 2nd to 3rd base and the side from 3rd base to home plate.

6. Now, set up the problem so that you can use the Pythagorean Theorem to find out how far the
catcher will have to throw the ball from home plate to 2nd base to the nearest foot.

a2 + b2 = c2  902 + 902 = x2  8100 + 8100 = x2
16,200 = x2  127.279 = x
Therefore, the catcher will have to throw approximately 127 feet.

7. How far will the third baseman have to throw to get to 2nd base. How can you determine this

The third baseman will have to throw the same distance of 127 feet.
Since the baseball diamond is a square, you can divide the square in half from the 3rd base
to the first base, creating 2 right triangles. The side lengths of these will be the same as in
the problem above. Therefore, the result will also be the same  approximately 127 feet.
Project AMP                          Dr. Antonio R. Quesada                 Director, Project AMP

Extension 1

1.) Complete the table for the special right triangle below. Express irrational values in

b
30º

a
c

60º

a              b               c              a2               b2               c2

1               3              2               1               3                4

3             3 3              6               9               27               36

5             5 3             10              25               75              100

6             6 3             12              36              108              144

2.) Do you notice a relationship between the lengths of the sides of this special right
triangle?
It appears that c is always twice as long as a. It also appears that b is always a factor
of the square root of 3.
Project AMP                          Dr. Antonio R. Quesada               Director, Project AMP

3.) Express the above relationship in terms of x.
If a = x, then c = 2x and b = x 3 .
4.) What conclusion can you make about 30°-60° right triangles?
The hypotenuse of a 30°-60° right triangle is always twice as long as the shortest side.
The other leg of a 30°-60° right triangle is always the product of the shortest side and   3.

Real world application – Locked Out!

José has locked himself out of his house. Fortunately, he did leave an
ladder, when fully extended safely, will extend to 24 feet. For optimal
safety reasons, he wants to maintain a 60-degree angle with the ground.
If he extends the ladder to 18 feet, (a) how far will the base have to be
away from the wall and (b)how high up on the wall will the ladder reach?

(a.) 9 feet
(b.) 9 3 feet
Project AMP                            Dr. Antonio R. Quesada              Director, Project AMP

Extension 2

1.) Complete the table for the special right triangle below. Express irrational values in

45º

a                     c

45º

b

a              B                c              a2             b2               c2

5              5             5 2               25             25               50

3              3             3 2                9             9                18

7              7             7 2               49             49               98

1              1                 2              1             1                2

2.) Do you notice a relationship between the lengths of the sides of this special right
triangle?
Yes, once we had our table filled in, we noticed that the two legs of the right triangle
are the same. Therefore, it could be called an isosceles right triangle. We also found that the
hypotenuses all contained factors of     2.

3.) Express the above relationship in terms of x.
If a = x, then b = x and c = x 2 .

4.) What conclusion can you make about 45-45 right triangles?
The two legs of any 45-45 right triangle will be the same. And, the hypotenuse of a 45-
45 right triangle is always a factor of 2 . We should, therefore, be able to find two missing
lengths of the sides of a 45-45 right triangle when we are given only one length.
Project AMP                      Dr. Antonio R. Quesada            Director, Project AMP

Real world application

6 feet
An Isosceles Right Triangle is the shape of the opening of a tent. What is the largest
sized object that can be put through the opening without touching the sides of the
opening?

9 feet2

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