pythagorean-theorem-distance-midpoint-2-2015-11-20.pptx - 1...

1

8th
Grade
Pythagorean Theorem, Distance &
Midpoint
2015-11-20
2

Table of Contents
Click on a topic to
•
Pythagorean Theorem
go to that
section
•
Distance Formula
• Midpoints
• Glossary
T
e
a
c
h
er
N
ot
e
s
3

Pythagorean Theorem
Click to return to
the table of contents
4

Pythagorean Theorem
Pythagorean theorem is used for right triangles. It was first
known in ancient Babylon and Egypt beginning about 1900 B.C.
However, it was not widely known until Pythagoras stated it.
Pythagoras lived during the 6th century B.C. on the island of
Samos in the Aegean Sea. He also lived in Egypt, Babylon, and
southern Italy. He was a philosopher and a teacher.
5

Legs
- Opposite the right angle
- Longest of the 3 sides
- 2 sides that form the right angle
click to reveal
c
a
b
Hypotenuse
click to reveal
click to reveal
Labels
for a right triangle
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In a right triangle, the sum of the squares of the
lengths of the legs ( a and b ) is equal to the
square of the length of the hypotenuse ( c ).
a
2
+ b
2
= c
2
Click on the links
below to see several
animations of the
proof
•
Water demo
•
Move slider to show c
2
•
Moving of squares
Pythagorean Theorem Proofs
7

Proof of the Pythagorean Theorem
Lab: Proof of the Pythagorean Theorem
Tea
che
r's
Not
es
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9
Follow-up Questions:
11. How is the area of the printed square on page 1 related to the
area of the printed square on page 2? Explain how you reached
that conclusion.
12. How are the 4 right triangles that you cut out at the beginning
of
M
at
h
Pr
ac
ti
ce
this lab related to each other? Explain how you reached
that conclusion.

10
Follow-up Questions (cont'd):
13. How are the areas that you found in question #6 & question
#10 related to each other? Explain how you reached that
conclusion.
14. What algebraic expression represents the side length of the
M
at
h
Pr
ac
tic
e
printed squares. Explain how you reached that conclusion.

11
Follow-up Questions (cont'd):
15. Multiply these side lengths together and simplify the expression.
16. Using the arrangement of the shapes from question #5, write an
M
at
h
P
ra
ct
ic
e
algebraic expression to represent the area of the entire figure.

Follow-up Questions (cont'd):
17. Set the expressions from question #15 & question #16 equal to
one another and simplify the equation.
Math Practice
12

a
2
+ b
2
= c
2
5
2
+ b
2
= 15
2
25 + b
2
= 225
-25 -
25
b
2
= 200
Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer
5 ft
15 ft
Pythagorean Theorem
13

9 in
18 in
a
2
+ b
2
= c
2
9
2
+ b
2
= 18
2
81 + b
2
= 324
-81 -81
b
2
= 243
Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer
Pythagorean Theorem
14

4 in
7 in
a
2
2
+ b = c
2
Missing Hypotenuse
Write Equation
4
2
+ 7
2
= c
2
S
16 + 49 = c
2
S
65 = c
2
A
F
L
ubstitute in numbers
quare numbers
dd
ind the Square Root &
abel Answer
Pythagorean Theorem
15

Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer
Missing Hypotenuse
Write Equation
Substitute in numbers
Square numbers
Add
Find the Square Root
Label Answer