Transcript Pythagorean Theorem

NGSSS
MA.8.G.2.4
The student will be able to:
Validate and apply Pythagorean
Theorem to find distances in real
world situations or between
points in the coordinate plane.
CCSS
8.G Understand and apply the
Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
7. Apply the Pythagorean Theorem to determine unknown side
lengths in right triangles in real-world and mathematical
problems in two and three dimensions.
8. Apply the Pythagorean Theorem to find the distance between
two points in a coordinate system.
The Pythagorean Theorem in HS:
The Pythagorean Theorem underlies several
formulas and identities that are memorized
by high school students. Related formulas include
•The Distance formula
•The Law of Cosines
•The equation of a Circle
•Some trigonometric identities.
Often, students memorize these formulas in
isolation, without being aware of their connection to
the Pythagorean Theorem.
What is a right triangle?
hypotenuse
leg
right angle
leg
• It is a triangle which has an angle that is
90 degrees.
• The two sides that make up the right angle
are called legs.
• The side opposite the right angle is the
hypotenuse.
The Pythagorean Theorem
In a right triangle, if a and b are the
measures of the legs and c is the
hypotenuse, then
2
a
+
2
b
=
2
c.
Note: The hypotenuse, c, is always
the longest side.
2
a
+
2
b
=
2
c
http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html
Proof of the Pythagorean Theorem
The image is the logo
from the Institute for
Mathematics & Education.
It provides us with an
elegant geometric “proof”
of the Pythagorean
Theorem.
Activity: How does this illustration prove the Pythagorean Theorem?
Proof of the Pythagorean Theorem
Given the red right
triangle, prove that the
area of the square of the
hypotenuse is equal to
the sum of the areas of
the squares of the two
legs.
The figure is formed from two large adjacent squares.
Each large square contains four congruent right triangles, one of which is
colored red.
Proof of the Pythagorean Theorem
The left square contains
two smaller squares.
The smallest square is
the result of the shorter
leg of the red right triangle.
The larger square is the result of the longer leg of the red right
triangle.
The largest square at the right is the result of the hypotenuse of the red
triangle.
Proof of the Pythagorean Theorem
Since both large squares
are equal, we can
subtract the four right
triangles from each
large square and still
have equal areas.
On the left are the squares of the two legs of the red right
triangle. On the right is the square of the hypotenuse.
Therefore, in a right triangle, the sum of the squares of the two
legs is equal to the square of the hypotenuse.
1. Find the length of the hypotenuse
122 + 162 = c2
144 + 256 = c2
400 = c2
12 in
16 in
Take the square root
of both sides.
400  c
2
C = ±20
The hypotenuse is 20 inches long.
2. Find the length of the hypotenuse
5 2 + 7 2 = c2
25 + 49 = c2
74 = c2
5 cm
7 cm
Take the square root of
both sides.
74  c
2
C » ±8.6
The hypotenuse is about 8.6 cm long.
3. Find the length of the hypotenuse
given that the legs of a right triangle
are 6 ft and 12 ft.
1.
2.
3.
4.
180 ft.
324 ft.
13.42 ft.
18 ft.
4. Find the length of the missing leg.
4 cm
10 cm
42 + b2 = 102
16 + b2 = 100
-16
-16
b2 = 84
b  84
2
b » ±9.2
The leg is about 9.2 cm long.
5. Find the length of the missing leg.
13 in
12 in
a2 + 122 = 132
a2 + 144 = 169
-144 -144
a2 = 25
a = 25
a = ±5
2
The leg is 5 inches long.
6. Find the length of the missing
side of a right triangle if one leg is 4
ft and the hypotenuse is 8 ft.
1.
2.
3.
4.
24 ft.
4 ft.
6.9 ft.
8.9 ft.
Application of Pythagorean Theorem
• The screen aspect ratio, or
the ratio of the width to the
length of a HDTV is 16:9.
The size of a television is
given by the diagonal distance across the
screen. If an HDTV is 41 inches wide, what is
its diagonal screen size?
• What are the dimensions of a 65 inch HDTV?
Application of Pythagorean Theorem
• A baseball diamond is a square with 90-foot
sides. What is the approximate distance the
catcher must throw from home to second
base?
The Converse of the Pythagorean Theorem
The Converse of the Pythagorean Theorem
A common application of the converse of the
Pythagorean Theorem is used by carpenters to
make sure a corner that they are constructing forms
a right angle. Here are the steps:
1.Starting at the corner, measure 3 units along
one direction and make a mark.
2. Measure 4 units along the other direction and make a mark.
3. Measure the distance between the marks.
4. If the length is equal to 5 units, then the corner forms a right angle (90°)
If the length is less than 5 units, then the corner is less than 90°
If the length is greater than 5 units, the corner is greater than 90°
Why? Since 32 + 42 = 52, then the triangle is a right triangle by the
converse of the Pythagorean Theorem.
5. The measures of three sides of a triangle
are given below. Determine whether the
triangle is a right triangle.
73 , 3, and 8
Which side is the longest?
The square root of 73 is about 8.5, therefore
it must be the hypotenuse.
Plug your information into the Pythagorean
Theorem. It doesn’t matter which number
is a or b.
Sides: 73 , 3, and 8
32 + 82 = ( 73 ) 2
9 + 64 = 73
73 = 73
Since this is true, the triangle is a
right triangle!! If it was not true, it
would not be a right triangle.
Three right triangles surround a shaded triangle;
together they form a rectangle measuring 12 units by 14
units. The figure below shows some of the dimensions
but is not drawn to scale.
14
7
12
5
9
5
Is the shaded
triangle a
right
triangle?
Provide
proof for
your answer.
14
7
245
12
15
50
9
5
5
No, the
shaded
triangle is
not a right
triangle.
2
15 + 50 ¹ 245
225 + 50 ¹ 245
2
275 ¹ 245
2
The Distance Formula
The distance formula is often
memorized in the square root
form with no connection to
previous learning.
Many students do not make the
connection that the distance
formula
is simply the Pythagorean
Theorem algebraically
manipulated
by solving for d, which is the
hypotenuse of a right triangle..
Deriving the Distance Formula
y 2 - y1
C
( x1, y 2 )
x 2 - x1
The Distance
from Point A to
Point B
would be equal
to the length of
the hypotenuse
of triangle
ABC.
Deriving the Distance Formula
a +b = c
2
2
( x1, y 2 )
2
2
x 2 - x1 + y 2 - y1 = c
y 2 - y1
C
2
x 2 - x1
Distance Formula =
c=±
2
( x2 - x1) + ( y2 - y1)
2
( x2 - x1) + ( y2 - y1)
2
2
2
Find the Distance Between:
1. Points A and B
2. Points B and C
3. Points A and C
Roland went on a hike to visit a cave in the mountains. To begin his
hike he faced west and hiked for 3 miles. Then he turned to the
south and traveled for 2 miles. After a water break Roland again
continued west for 4 miles. Turning North he continued for 3
miles. Next Roland turned left for 2 miles, and then he took a right
and continued on his hike for a final 6 miles until he discovered the
location of the cave.
As “the crow flys”,
how far is the cave
from where Roland
started his hike?
West 3 miles
South 2 miles
West 4 miles
North 3 miles
Left 2 miles
Right 6 miles
9 2 + 72 = c 2
81+ 49 = c 2
130 = c 2
130 = c 2
c » 11.4 miles
9
7
Pythagorean Theorem in the 3rd Dimension
What is the longest curtain rod you can fit in this box?
Note: Fishing poles only come in increments of tenth of a foot.
2 +4 =c
2
2
4 +16 = c
3
20 = c
?
?2
?20
2
c = ± 20
2
2
Pythagorean Theorem in the 3rd Dimension
What is the longest curtain rod you can fit in this box?
Note: Fishing poles only come in increments of tenth of a foot.
2
3 + 20 = x
2
9 + 20 = x
3
2
?
20
29 = x
2
2
x = ± 29
» 5.3851
The longest rod is 5.3 feet
2
Pythagorean Theorem in the 3rd Dimension
Derive a formula that will always work to find the diagonal of any
rectangular prism.
d = l +w +h
2
2
2