Download THE PYTHAGOREAN THEOREM

Lesson 7-2
The Pythagorean
Theorem
Lesson 7-2: The Pythagorean Theorem
1
Anatomy of a right triangle
• The hypotenuse of a right triangle is the
longest side. It is opposite the right angle.
• The other two sides are legs. They form the
right angle.
leg
hypotenuse
leg
Lesson 7-2: The Pythagorean Theorem
2
The Pythagorean Theorem
1. Draw a right triangle with lengths
a, b and c. (c the hypotenuse)
2. Draw a square on each side of
the triangle.
3. What is the area of each square?
a2
a
b2
b
c
c2
Lesson 7-2: The Pythagorean Theorem
The Pythagorean Theorem
a2
a
The Pythagorean
Theorem says
a2 + b2 = c2
b2
b
c
c2
Lesson 7-2: The Pythagorean Theorem
Proofs of the Pythagorean Theorem
Proof 1
Proof 2
Proof 3
Lesson 7-2: The Pythagorean Theorem
5
The Pythagorean Theorem
If a triangle is a right triangle, with leg
lengths a and b and hypotenuse c,
then a2 + b2 = c2
c
a
c is the
length of the
hypotenuse!
b
Lesson 7-2: The Pythagorean Theorem
6
The Pythagorean Theorem
If a triangle is a right triangle,
then leg2 + leg2 = hyp2
leg
hyp
leg
Lesson 7-2: The Pythagorean Theorem
7
Example
In the following figure if a = 3 and b = 4, Find c.
leg2 + leg2 = hyp2
32 + 4 2 = C 2
c
a
9 + 16 = C2
25 = C2
25 
2
C
5 = C
b
Lesson 7-2: The Pythagorean Theorem
8
Pythagorean Theorem : Examples
1. a = 8, b = 15, Find c
c = 17
2. a = 7, b = 24, Find c
c = 25
3. a = 9, b = 40, Find c
c = 41
4. a = 10, b = 24, Find c
c = 26
5.
c = 10
a = 6, b = 8, Find c
a
Lesson 7-2: The Pythagorean Theorem
c
b
9
Finding the legs of a right triangle:
In the following figure if b = 5 and c = 13, Find a.
leg2 + leg2 = hyp2
a2 +52 = 132
c
a
b
a2 + 25 = 169
-25 -25
a2
= 144
a2 = 144
a = 12
Lesson 7-2: The Pythagorean Theorem
10
More Examples:
1)
2)
3)
4)
5)
6)
7)
8)
a=8, c =10 , Find b
a=15, c=17 , Find b
b =10, c=26 , Find a
a=15, b=20, Find c
a =12, c=16, Find b
b =5, c=10, Find a
a =6, b =8, Find c
a=11, c=21, Find b
b=6
b=8
a = 24
c = 25
b  112
a = 8.7
c = 10
c
a
b  320  8 5
b
Lesson 7-2: The Pythagorean Theorem
11
A Little More Triangle Anatomy
• The altitude of a triangle is a segment from a
vertex of the triangle perpendicular to the
opposite side.
altitude
Lesson 7-2: The Pythagorean Theorem
12
Altitude - Special Segment of Triangle
Definition: a segment from a vertex of a triangle perpendicularBto the
segment that contains the opposite side.
C
AF , BE , DC are the altitudes of the triangle.
In a right triangle, two of the
altitudes are the legs of the triangle.
B
A
K
E
A
A
D
F
F
AB, AD, AF  altitudes of right
F
I
B
D
D
In an obtuse triangle, two of the altitudes
are outside of the triangle.
BI , DK , AF  altitudes of obtuse
Lesson 3-1: Triangle Fundamentals
13
Example:
• An altitude is drawn to the side of an equilateral
triangle with side lengths 10 inches. What is the
length of the altitude?
h2 + 52 = 102
h2 = 75
h=
10 in
h
10 in
75
 25  3
 5 3 in
?5
10 in
Lesson 7-2: The Pythagorean Theorem
14
The Pythagorean Theorem – in Review
Pythagorean Theorem:
If a triangle is a right triangle, with side
lengths a, b and c (c the hypotenuse,)
c
a
then a2 + b2 = c2
What is the
converse?
b
Lesson 7-2: The Pythagorean Theorem
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The Converse of the Pythagorean Theorem
If, a2 + b2 = c2,
then the triangle is a right triangle.
c
a
C is the
LONGEST side!
b
Lesson 7-2: The Pythagorean Theorem
16
Given the lengths of three sides,
how do you know if you have a
right triangle?
Given a = 6, b=8, and c=10, describe the triangle.
Compare a2 + b2 and c2:
c
a
b
a 2 + b2
62 + 82
36 + 64
100
=
=
=
=
c2
102
100
100
Since 100 = 100, this is a right triangle.
Lesson 7-2: The Pythagorean Theorem
17
The Contrapositive of the Pythagorean Theorem
If a2 + b2 c2
then the triangle is NOT a right triangle.
c
a
b
Lesson 7-2: The Pythagorean Theorem
What if
a2 + b2 c2
?
18
The Contrapositive of the Pythagorean Theorem
If a2 + b2  c2
then either, a2 + b2 > c2 or a2 + b2 < c2
c
a
b
Lesson 7-2: The Pythagorean Theorem
What if
a2 + b2 c2
?
19
The Converse of the Pythagorean Theorem
If a2 + b2 > c2 , then
the triangle is acute.
c
a
The longest
side is too
short!
b
Lesson 7-2: The Pythagorean Theorem
20
The Converse of the Pythagorean Theorem
If a2 + b2 < c2 , then
the triangle is obtuse.
a
b
The longest
side is too
long!
c
Lesson 7-2: The Pythagorean Theorem
21
Given the lengths of three sides,
how do you know if you have a
right triangle?
Given a = 4, b = 5, and c =6, describe the triangle.
Compare a2 + b2 and c2:
a
c
b
a 2 + b2
42 + 52
16 + 25
41
>
>
>
>
c2
62
36
36
Since 41 > 36, this is an acute triangle.
Lesson 7-2: The Pythagorean Theorem
22
Given the lengths of three sides,
how do you know if you have a
right triangle?
Given a = 4, b = 6, and c = 8, describe the triangle.
Compare a2 + b2 and c2:
b
a
c
a 2 + b2
42 + 62
16 + 36
52
<
<
<
<
c2
82
64
64
Since 52 < 64, this is an obtuse triangle.
Lesson 7-2: The Pythagorean Theorem
23
Describe the following triangles as acute, right, or obtuse
right 1) 9, 40, 41
obtuse 2) 15, 20, 10
obtuse 3) 2, 5, 6
4) 12,16, 20
right
5) 14,12,11
acute
6) 2, 4, 3
obtuse
7) 1, 7, 7
acute
8) 90,150, 120
c
a
right
Lesson 7-2: The Pythagorean Theorem
b
24
Application
The Distance
Formula
The Pythagorean Theorem
• For a right triangle with legs of
length a and b and hypotenuse of
length c,
c  a b
2
or
2
2
c  a b
2
2
The x-axis
• Start with a horizontal number line which we will call the xaxis.
• We know how to measure the distance between two points
on a number line.
x
Take the absolute value of the difference:
│a – b │
│ – 4 – 9 │= │ – 13 │ = 13
The y-axis
• Add a vertical number line which we will call the y-axis.
• Note that we can measure the distance between two
points on this number line also.
y
x
The Coordinate Plane
We call the x-axis together with the y-axis the coordinate plane.
y
x
Coordinates / Ordered Pair
• Coordinates –
numbers that identify
the position of a point
• Ordered Pair – a pair
of numbers (xcoordinate, y-coordinate)
identifying a point’s
position
Identify some coordinates and ordered pairs in the diagram.
Diagram is from the website www.ezgeometry.com .
Finding Distance in
The Coordinate Plane
We can find the distance between any two points in the coordinate plane
by using the Ruler Postulate AND the Pythagorean Theorem.
y
?
x
Finding Distance in
The Coordinate Plane cont.
First, draw a right triangle.
y
?
x
Finding Distance in
The Coordinate Plane cont.
Next, find the lengths of the two legs.
•First, the horizontal leg:
│(– 4) – 8│= │– 12│ = 12
y
–4
12
?
8
x
Finding Distance in
The Coordinate Plane cont.
So the horizontal leg is 12 units long.
•Now find the length of the vertical leg:
│3 – (– 2)│= │ 5 │ = 5
y
3
5
?
x
–2
12
Finding Distance in
The Coordinate Plane cont.
Here is what we know so far.
Since this is a right triangle, we use the Pythagorean Theorem.
c  5  12
2
2
y
The distance is 13 units.
 169
 13
?
13
12
5
x
The Distance Formula
Instead of drawing a right triangle and using the Pythagorean Theorem, we can
use the following formula:
distance =
 x2  x1    y2  y1 
2
2
where (x1, y1) and (x2, y2) are the ordered pairs corresponding to the two points.
So let’s go back to the example.
Example
Find the distance between these two points.
Solution:
First
: Find the coordinates of each point.
y
(8, 3)
–4
3
?
x
–2
(– 4, – 2)
8
Example
Find the distance between these two points.
Solution:
First: Find the coordinates of each point.
(x1, y1) = (-4, -2)
y
(x2, y2) = (8, 3)
(8, 3)
?
x
(– 4, – 2)
Example cont.
Solution cont.
Then: Since the ordered pairs are
(x1, y1) = (-4, -2) and (x2, y2) = (8, 3)
Plug in x1 = -4, y1 = -2, x2 = 8 and y2 = 3 into
distance
=
 x2  x1    y2  y1 
2
2
=
8  4  3  2
2
=
12  5
2
=
13
2
2