The Pythagorean Theorem
Problem 3.1
LEG
The longest side of a right triangle is the side
opposite the right angle. We call this side the
HYPOTENUSE of the triangle. The other two
sides are called LEGS.
LEG
Consider a right triangle with legs that each
have a length of 1. Suppose you draw squares
on the hypotenuse and the legs of the triangle.
How are the areas of these three squares related?
12
10
Draw the required squares on your dot paper and
complete the chart. Look for patterns.
18
16
14
Length of
leg 1
1
1
2
1
2
3
3
Length of
Leg 2
1
2
2
3
3
3
4
Area of
square on
leg 1
1
1
4
1
4
9
9
Area of
square on
leg 2
1
4
4
9
9
9
16
Area of
square on
hypotenuse
2
5
8
10
13
18
25
hypotenuse
length
1.4142
2.2361
2.8284
3.1623
3.6056
4.2426
5
Draw the required squares on your dot paper and
complete the chart. Look for patterns.
18
16
14
Length of
leg 1
1
1
2
1
2
3
3
Length of
Leg 2
1
2
2
3
3
3
4
Area of
square on
leg 1
1
1
4
1
4
9
9
Area of
square on
leg 2
1
4
4
9
9
9
16
Area of
square on
hypotenuse
2
5
8
10
13
18
25
hypotenuse
length
1.4142
2.2361
2.8284
3.1623
3.6056
4.2426
5
Draw the required squares on your dot paper and
complete the chart. Look for patterns.
18
16
14
Length of
leg 1
1
1
2
1
2
3
3
Length of
Leg 2
1
2
2
3
3
3
4
Area of
square on
leg 1
1
1
4
1
4
9
9
Area of
square on
leg 2
1
4
4
9
9
9
16
Area of
square on
hypotenuse
2
5
8
10
13
18
25
hypotenuse
length
1.4142
2.2361
2.8284
3.1623
3.6056
4.2426
5
Draw the required squares on your dot paper and
complete the chart. Look for patterns.
18
16
14
12
Length of
leg 1
1
1
2
1
2
3
3
Length of
Leg 2
1
2
2
3
3
3
4
Area of
square on
leg 1
1
1
4
1
4
9
9
Area of
square on
leg 2
1
4
4
9
9
9
16
Area of
square on
hypotenuse
2
5
8
10
13
18
25
hypotenuse
length
1.4142
2.2361
2.8284
3.1623
3.6056
4.2426
5
Draw the required squares on your dot paper and
complete the chart. Look for patterns.
Length of
leg 1
1
1
2
1
2
3
3
Length of
Leg 2
1
2
2
3
3
3
4
Area of
square on
leg 1
1
1
4
1
4
9
9
Area of
square on
leg 2
1
4
4
9
9
9
16
Area of
square on
hypotenuse
2
5
8
10
13
18
25
hypotenuse
length
1.4142
2.2361
2.8284
3.1623
3.6056
4.2426
5
D
E
B
c
a
A
Length of
leg 1
1
1
2
1
2
3
3
Length of
Leg 2
1
2
2
3
3
3
4
Area of
square on
leg 1
1
1
4
1
4
9
9
b
C
Area of
square on
leg 2
1
4
4
9
9
9
16
Area of
square on
hypotenuse
2
5
8
10
13
18
25
hypotenuse
length
1.4142
2.2361
2.8284
3.1623
3.6056
4.2426
5
C. Look for patterns in the relationship among the
areas of the three squares drawn for each triangle.
Use the pattern you discover to make a conjecture
(hypothesis) about the relationship among the areas.
Length of
leg 1
1
1
2
1
2
3
3
Length of
Leg 2
1
2
2
3
3
3
4
Area of
square on
leg 1
1
1
4
1
4
9
9
Area of
square on
leg 2
1
4
4
9
9
9
16
Area of
square on
hypotenuse
2
5
8
10
13
18
25
hypotenuse
length
1.4142
2.2361
2.8284
3.1623
3.6056
4.2426
5
Area of Area of
Area of
Leg 1 + Leg 2 = Hypotenuse
square Square
square
D. Draw a right triangle with side lengths that are different from those
given in the table. Use your triangle to test your conjecture from part C.
Go to Geometer’s SketchPad
Now that I know the area of the Hypotenuse
Square, I can I find the one thing that I don’t know
about a right triangle?
D
B
40 sq
units
c
6a
E
A
4 sq
units
2
b
C
36 sq
units
3.1 Follow Up: Record the length of the hypotenuse
using the square root method. Approximate your
answer to the nearest hundredth.
Length of
leg 1
1
1
2
1
2
3
3
Length of
Leg 2
1
2
2
3
3
3
4
Area of
square on
leg 1
1
1
4
1
4
9
9
Area of
square on
leg 2
1
4
4
9
9
9
16
Area of
square on
hypotenuse
2
5
8
10
13
18
25
hypotenuse
length
1.41
2.24
2.83
3.16
3.61
4.24
5.00
Here is the importance of what you just learned:
You were given the lengths of the two LEGS and were asked to
find the HYPOTENUSE.
You found that if you found the areas of the two squares formed
from legs and added them up, it equaled the area of the square
formed from the hypotenuse.
Then you found the length of the hypotenuse by taking the square
root of that larger area.
Try these two problems. Find the length of the hypotenuse for each.
Leg 4 .
Leg
2 .
Hypotenuse
.
Leg 5 .
Leg
6 .
Hypotenuse
.
Leg 4 .
4 x 4 = 16
Leg 5 .
5 x 5 = 25
Leg
2 .
2x2=4
Leg
6 .
6 x 6 = 36
Hypotenuse
4.47 .
16 + 4 = 20
20 = 4.47
Hypotenuse
7.81 .
25 + 36 = 61 61 = 7.81
a
Pythagoras wrote his theorem as:
It only matters that
2
2
2
a +b =c
“c” is in the spot of
b
the hypotenuse in
the formula. It
does not matter
which leg you
make “a” and
which leg you
make “b”.
You just proved this:
2
a
+
D
B
c2
c
a
E
A
b
b2
C
a2
2
b
=
2
c
If you knew the hypotenuse and
one of the legs could you find the
other?
Leg 9 .
Leg
? .
Hypotenuse
15
a2 + b2 = c2
92 + b2 = 152
81+ b2 = 225
b2 = 144
b2 = 144 = 12
.