Transcript 20 1 Pythag Theorem

20-1: Pythagorean Theorem
Objectives:
1. To discover, use,
and prove the
Pythagorean
Theorem
2. To use Pythagorean
Triples to find
quickly find a
missing side length
in a right triangle
Assignment:
• P. 285: 10-15
• P. 289: 1-10
• Challenge Problems
Objective 1
You will be able to discover, use,
and prove the Pythagorean Theorem
Example 1
1. What is the area of a
square with a side
length of 4 inches? x
inches?
x
2. What is the side
length of a square
with an area of 25
in2? x in2?
x
Parts of a Right Triangle
Which segment is the longest in any right
triangle?
Tangram Activity
Take out your 7
tangrams, or if you
don’t have them, cut
out some new ones.
Step 1: In the center
of a new sheet of
paper, trace one of
the large right
triangles.
Tangram Activity
Step 2: Use all of your
tangram pieces to
assemble two squares
along the legs of your
traced triangle so that
the length of each leg is
equal to the side length
of the square. Trace all
of the pieces.
Tangram Activity
Step 3: Find the area of
each assembled square.
How does your answer
relate to the length of
each leg?
Tangram Activity
Step 4: Use all of your
tangram pieces to
assemble a square along
the hypotenuse of your
traced triangle so that the
side length of the square
is equal to the length of
the hypotenuse. Trace
all of the pieces.
Tangram Activity
Step 5: Find the area of the
large square. How does
your answer relate to the
length of the
hypotenuse?
Tangram Activity
Step 6: How do the
areas of each
square relate to
each other? How
do the areas of the
squares relate the
legs and
hypotenuse of your
traced triangle?
Investigation 1
Use the Geometer’s
Sketchpad Activity
to investigate the
relationship
between the legs of
a right triangle and
its hypotenuse.
Click on the picture
to start.
The Pythagorean Theorem
In a right triangle, the
square of the length of
the hypotenuse is
equal to the sum of the
squares of the lengths
of the legs.
The Pythagorean Theorem
In a right triangle, the
square of the length of
the hypotenuse is
equal to the sum of the
squares of the lengths
of the legs.
c2
a2
b2
The Pythagorean Theorem
In a right triangle, the
square of the length of
the hypotenuse is
equal to the sum of the
squares of the lengths
of the legs.
Example 2
The triangle below is definitely not a right
triangle. Does the Pythagorean Theorem
A
work on it?
m BA = 11.85 cm
m AC = 7.62 cm
B
37
69
m CB = 12.15 cm
C
Example 3
How high up on the
wall will a twentyfoot ladder reach if
the foot of the
ladder is placed five
feet from the wall?
Example 4: SAT
In figure shown, what
is the length of RS?
S
3
T
7
R
Pythagorean Theorem: Proof
Use the properties of similar right triangles to
show that given a right triangle, 𝑎2 + 𝑏 2 = 𝑐 2 .
Example 5
What is the area of the large square?
Example 6
Find the area of the
triangle.
Objective 2
You will be able to use Pythagorean
Triples to find quickly find a missing
side length in a right triangle
Pythagorean Triples
Three whole numbers that work in the
Pythagorean formulas are called
Pythagorean Triples.
Example 7
What happens if you add the same length to
each side of a right triangle? Do you still
get another right triangle?
Example 8
What happens if you multiply all the side
lengths of a right triangle by the same
number? Do you get another right
triangle?
Pythagorean Multiples
Pythagorean Multiples Conjecture:
If you multiply the lengths of all three sides of
any right triangle by the same number,
then the resulting triangle is a right triangle.
In other words, if a2 + b2 = c2, then
(an)2 + (bn)2 = (cn)2.
Pythagorean Triples
Pythagorean Triples
Primitive Pythagorean Triples
A set of Pythagorean triples is considered a
primitive Pythagorean triple if the
numbers are relatively prime; that is, if they
have no common factors other than 1.
3-4-5
5-12-13
7-24-25
8-15-17
9-40-41
11-60-61
12-35-37
13-84-85
16-63-65
20-21-29
28-45-53
33-56-65
36-77-85
39-80-89
48-55-73
65-72-97
Example 9
Find the length of one leg of a right triangle
with a hypotenuse of 35 cm and a leg of 28
cm.
Example 10
Use Pythagorean Triples to find each
missing side length.
Example 11
A 25-foot ladder is placed against a building.
The bottom of the ladder is 7 feet from the
building. If the top of the ladder slips down
4 feet, how many feet will the bottom slip
out?
20-1: Pythagorean Theorem
Objectives:
1. To discover, use,
and prove the
Pythagorean
Theorem
2. To use Pythagorean
Triples to find
quickly find a
missing side length
in a right triangle
Assignment:
• P. 285: 10-15
• P. 289: 1-10
• Challenge Problems