Transcript Geo Ch 8-2 – The Pythagorean Theorem

8 – 2: The Pythagorean
Theorem
Textbook pp. 440 - 446
• Use the Pythagorean Theorem and its converse.
• Pythagorean triple
Standard 12.0 Students find and use measures of sides
and of interior and exterior angles of triangles and polygons
to classify figures and solve problems. (Key)
Standard 14.0 Students prove the Pythagorean
theorem. (Key)
Standard 15.0 Students use the Pythagorean theorem
to determine distance and find missing lengths of sides
of right triangles.
Pythagorean Theorem
• In a right triangle, the sum of the squares
of the measures of the legs equals the
square of the measure of the hypotenuse.
a2 + b 2 = c 2
Click here for the
Pythagorean
Proof
Find x. Round your answer to the nearest tenth.
A. 17
B. 12.7
0%
C. 11.5
D. 13.2
1.
2.
3.
4.
A
B
C
D
A
B
C
D
Verify a Triangle is a Right Triangle
COORDINATE GEOMETRY Verify that ΔABC is a
right triangle.
Use distance
formula on all
3 sides then
the
Pythagorean
theorem.
Verify a Triangle is a Right Triangle
COORDINATE GEOMETRY Verify that ΔABC is a
right triangle.
10
2
2  10
2
2

104
6 4
2
4 6
4
2
2


52
 

52
6
4
2 ?
52

52
6
2
2

2
104
52  52  104
COORDINATE GEOMETRY
Is ΔRST a right triangle?
A.
yes
B.
no
1.
2.
3.
cannot be
determined
0%
C
0%
B
0%
A
C.
A
B
C
Pythagorean Triples
A. Determine whether 9, 12, and 15 are the sides of
a right triangle. Then state whether they form a
Pythagorean triple.
Since the measure of the longest side is 15, 15 must be c.
Let a and b be 9 and 12.
Pythagorean Theorem
Simplify.
Add.
Homework
Chapter 8-2
 Pg 444:
#1 – 3, 6 – 26
W
=L
x
A
y = mx + b
3x
=
+5
A
14
= p
r
2
M ath Zone
The Pythagorean Theorem: a 2  b 2  c 2
(Area of green square) + (Area of red square) = Area of the blue square
1. We start with half the red square, which has
Area = ½ base x height
3. We rotate this triangle, which does
not change its area.
4. We mark the base and height for
this triangle.
height
2. We move one vertex while maintaining the
base & height, so that the area remains the
same. This is called a SHEAR.
base
b
2
c
b
c
a
a
2
2
W
=L
x
A
y = mx + b
3x
=
+5
A
14
= p
r
2
M ath Zone
The Pythagorean Theorem: a 2  b 2  c 2
(Area of green square) + (Area of red square) = Area of the blue square
1. We start with half the red square, which has
Area = ½ base x height
2. We move one vertex while maintaining the
base & height, so that the area remains the
same. This is called a SHEAR.
3. We rotate this triangle, which does
not change its area.
c
b
2
4. We mark the base and height for
this triangle.
5. We now do a shear on this triangle,
keeping the same area.
Remember that this pink triangle is half the red square.
a
2
2
W
=L
x
A
y = mx + b
3x
=
+5
A
14
= p
r
2
M ath Zone
The Pythagorean Theorem: a 2  b 2  c 2
(Area of green square) + (Area of red square) = Area of the blue square
6. The other half of the red square has the same area as
this pink triangle, so if we copy and rotate it, we get this.
So, together these two pink triangles have the same
area as the red square.
7. We now take half of the green square and
transform it the same way.
We end up with this triangle, which is
half of the green square.
cc
2
b
2
2
Rotate
8. The other half of the green square
would give us this.
Shear
9. Together, they have they same
area as the green square.
So, we have shown that the red & green squares
together have the same area as the blue square.
a
2
Shear
W
=L
x
3x
=
+5
A
A
y = mx + b
14
= p
r
2
M ath Zone
The Pythagorean Theorem: a 2  b 2  c 2
We’ve Proven
the
Pythagorean
Theorem
(Area of green square) + (Area of red square) = Area of the blue square
6. The other half of the red square has the same area as
this pink triangle, so if we copy and rotate it, we get this.
So, together these two pink triangles have the same
area as the red square.
7. We now take half of the green square and
transform it the same way.
cc
2
We end up with this triangle, which is
half of the green square.
8. The other half of the green square
would give us this.
b
We’ve PROVEN the Pythagorean Theorem!
Shear
2
Rotate
9. Together, they have they same
area as the green square.
So, we have shown that the red & green squares
together have the same area as the blue square.
2
Shear
a
2
(click to return)