Use the Pythagorean Theorem to solve problems

1. The Pythagorean Theorem
The student is able to (I can):
• Use the Pythagorean Theorem to solve problems.

2. The Pythagorean TheoremPythagorean TheoremPythagorean TheoremPythagorean Theorem (a2 + b2 = c2) states the
relationship between the sides of a right triangle. Although it
was named for Pythagoras (circa 500 B.C.), this relationship
was actually known to earlier people, including the
Babylonians, Egyptians, and the Chinese.
A Babylonian tablet from 1800 B.C. that is
presumed to be listing sides of right triangles.

3. Proving the Pythagorean Theorem
While there are many known proofs of the theorem, one
method we will examine today is a graphical one.
1. Cut out the triangles and the square.
2. Arrange them as shown. What is the area of this shape?
a
b
c
a
b
c
b
a
b
c
a c
b–a
b–a
b–a
b–a

4. Proving the Pythagorean Theorem
While there are many known proofs of the theorem, one
method we will examine today is a graphical one.
1. Cut out the triangles and the square.
2. Arrange them as shown. What is the area of this shape?
a
b
c
a
b
c
b
a
b
c
a c
b–a
b–a
b–a
b–a
aaaa2222
bbbb2222
2 2
a b

5. 3. Rearrange them to form a square with side length c.
What is the area of this square?
a b
c
c
a
c
a
bb
b
c
a
c

6. 3. Rearrange them to form a square with side length c.
What is the area of this square?
a b
c
c
a
cccc2222 c
a
bb
b
c
a
c
cccc2222

7. Conclusion
Since we are using the same triangles and square for both
shapes, this means that the areas are the same, therefore:
The Pythagorean Theorem allows us to find an unknown side
of a right triangle if we know the other two sides.
Remember: the hypotenuseRemember: the hypotenuseRemember: the hypotenuseRemember: the hypotenuse (opposite the right angle) is(opposite the right angle) is(opposite the right angle) is(opposite the right angle) is
2 2 2
a b c 
Remember: the hypotenuseRemember: the hypotenuseRemember: the hypotenuseRemember: the hypotenuse (opposite the right angle) is(opposite the right angle) is(opposite the right angle) is(opposite the right angle) is
alwaysalwaysalwaysalways cccc....
hypotenuse
right
angle

8. Example: Find the value of x.
x
12
13

9. Example: Find the value of x.
x
12
13
 2 2 2
12 13x

10. Example: Find the value of x.
x
12
13
 2 2 2
12 13x
 

2
2
144 169
25
x
x 
 
2
25
25 5
x
x

11. Square Roots
• When we are taking the square root of a number, we will
not always get a whole number answer.
• If your answer is not a whole number, then the number
your calculator gives you is a decimal approximationapproximationapproximationapproximation. This
is an irrational number, like , which goes on forever and
does not repeat.
• If I ask for an exact answerexact answerexact answerexact answer, I do notnotnotnot want a decimal – I• If I ask for an exact answerexact answerexact answerexact answer, I do notnotnotnot want a decimal – I
want you to leave it as a radicalradicalradicalradical.

12. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1. 2
6
x
2.
x x-2
4

13. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2 2 2
2 6 x 
2
4 36 x 
2
6
x
2.
x x-2
4

14. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2 2 2
2 6 x 
2
4 36 x 
2
40 x
40x 
2
6
x
2.
x x-2
4

15. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2 2 2
2 6 x 
2
4 36 x 
2
40 x
40x 
2
6
x
2.
2 2 2
4 ( 2)x x  
x x-2
4

16. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2 2 2
2 6 x 
2
4 36 x 
2
40 x
40x 
2
6
x
2.
2 2 2
4 ( 2)x x   xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2 2
16 4 4x x x   
x x-2
4

17. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2 2 2
2 6 x 
2
4 36 x 
2
40 x
40x 
2
6
x
2.
2 2 2
4 ( 2)x x   xxxx ----2222
xxxx x2 -2x
----2222 -2x 4
2 2
16 4 4x x x   
20 – 4x = 0
20 = 4x
x = 5
x x-2
4