Transcript Document 7206401
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 leg hypotenuse
Lesson 7-2: The Pythagorean Theorem 2
b 2
The Pythagorean Theorem
a 2 a
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?
b c c 2
Lesson 7-2: The Pythagorean Theorem
b 2
The Pythagorean Theorem
a 2 a
The Pythagorean Theorem says
a 2 + b 2 = c 2 b c c 2
Lesson 7-2: The Pythagorean Theorem
Proofs of the Pythagorean Theorem
Proof 1 Proof 2 Proof 3 Lesson 7-2: The Pythagorean Theorem 5
a The Pythagorean Theorem
If a triangle is a right triangle, with leg lengths a and b and hypotenuse c, then
a 2 + b 2 = c 2 c
c is the length of the hypotenuse!
b
Lesson 7-2: The Pythagorean Theorem 6
leg The Pythagorean Theorem
If a triangle is a right triangle, then
leg 2 + leg 2 = hyp 2 hyp leg
Lesson 7-2: The Pythagorean Theorem 7
Example
In the following figure if a = 3 and b = 4, Find c.
leg 2 + leg 2 = hyp 2 3 2 + 4 2 = C 2
a c
9 + 16 = C 2 25 = C 2 25 5 = C
C
2
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 3.
a = 9, b = 40, Find c
c = 25 c = 41 a
4.
a = 10, b = 24, Find c
c = 26
5.
a = 6, b = 8, Find c
c = 10
Lesson 7-2: The Pythagorean Theorem
b c
9
Finding the legs of a right triangle: In the following figure if b = 5 and c = 13, Find a.
leg 2 + leg 2 = hyp 2
a
a 2 +5 2 = 13 2
b c
a 2 + 25 = 169 -25 -25 a 2 = 144 a 2 = 144 a = 12 Lesson 7-2: The Pythagorean Theorem 10
More Examples:
1) a=8, c =10 , Find b 2) a=15, c=17 , Find b 3) b =10, c=26 , Find a 4) a=15, b=20, Find c 5) a =12, c=16, Find b 6) b =5, c=10, Find a 7) a =6, b =8, Find c 8) a=11, c=21, Find b
b = 6 b = 8 a = 24 c = 25
b
112
a = 8.7
b
c = 10
320 8 5
a b c
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 perpendicular to the segment that contains the opposite side.
B C , .
B F In a
right triangle
, two of the altitudes are the legs of the triangle.
B ,
altitudes of right
F A E D F I A K D A In an
obtuse triangle ,
D two of the altitudes are outside of the triangle.
,
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?
h 2 + 5 2 = 10 2 h 2 = 75 10 in h 10 in h =
75 5 3 in
?5
10 in
Lesson 7-2: The Pythagorean Theorem 14
a 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
then
a 2 + b 2 = c 2
What is the converse?
b
Lesson 7-2: The Pythagorean Theorem 15
a The Converse of the Pythagorean Theorem
If,
a 2 + b 2 = c 2
, then the triangle is a right triangle.
c
C is the LONGEST side!
b
Lesson 7-2: The Pythagorean Theorem 16
a 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 a 2 + b 2
c
and c 2: a 2 + b = c 2 6 2 + 8 36 + 64 100 = = = 10 100 100 2
b
Since 100 = 100, this is a right triangle.
Lesson 7-2: The Pythagorean Theorem 17
The Contrapositive of the Pythagorean Theorem a
If
a 2 + b 2
c 2
then the triangle is NOT a right triangle.
c
What if
a 2 + b 2
c 2 ?
b
Lesson 7-2: The Pythagorean Theorem 18
The Contrapositive of the Pythagorean Theorem a
If
a 2 + b 2
c 2
then either,
a 2 + b 2 > c 2
or
a 2 + b 2 < c 2 c
What if
a 2 + b 2
c 2 ?
b
Lesson 7-2: The Pythagorean Theorem 19
The Converse of the Pythagorean Theorem a b
If
a 2 + b 2 > c 2
, then
the triangle is acute.
c
Lesson 7-2: The Pythagorean Theorem The longest side is too short!
20
The Converse of the Pythagorean Theorem
If
a 2 + b 2 < c 2
, then
the triangle is obtuse.
a c b
Lesson 7-2: The Pythagorean Theorem The longest side is too long!
21
a 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 a 2 + b 2
c
and c 2: a 2 + b 2 c 2 4 2 + 5 16 + 25 41 > > > > 6 2 36 36
b
Since 41 > 36, this is an acute triangle.
Lesson 7-2: The Pythagorean Theorem 22
a 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 a 2 + b 2
b
and c 2: a 2 + b 2 c 2 4 2 + 6 16 + 36 52 < < < < 8 2 64 64
c
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
right a
Lesson 7-2: The Pythagorean Theorem
b c
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 2 or c a 2 b 2 a 2 b 2
The x-axis
• • Start with a horizontal number line which we will call the x axis.
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
(x coordinate, 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 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
5 3 – 2 ?
12
x
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.
y
c 13 5 2 169 12 2 The distance is 13 units .
5
x
12
The Distance Formula
Instead of drawing a right triangle and using the Pythagorean Theorem, we can use the following formula: distance = x 2 x 1 y 2 y 1 2 where (x 1 , y 1 ) and (x 2 , y 2 ) 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
– 4 3 ?
– 2 ( – 4, – 2) (8, 3) 8
x
Example
Find the distance between these two points.
Solution
:
First
: Find the coordinates of each point.
(x 1 , y 1 ) = (-4, -2) (x 2 , y 2 ) = (8, 3)
y
?
(8, 3)
x
( – 4, – 2)
Example cont.
Solution cont.
Then
: Since the ordered pairs are (x 1 , y 1 ) = (-4, -2) and (x 2 , y 2 ) = (8, 3) Plug in x 1 = -4, y 1 = -2, x 2 = 8 and y 2 = 3 into distance = x 2 x 1 = 8 4 = y 2 y 1 2 3 2 2 2 = 13