4.8d Discover the Pythagorean Theorem
Holt Course 3
Name: ______________________________
Earlier you found areas of figures drawn on dot paper. Some of these figures were squares,
either “upright” and “tilted” squares. When you found those areas, you may have used a method
that involved triangles, specifically right triangles. Recall that a right triangle is a triangle that
has a right, or a 90º, angle. The right angle is often marked with a small square.
The right angle is formed when two of the sides meet at exactly 90º. The two sides that form this
90º angle are called the “legs”. The third side, which is always the longest side of a right
triangle, is called the “hypotenuse.” This longest side is always opposite the right angle.
Analyze the right triangles below to make sure you understand the difference between the legs
and the hypotenuse of a right triangle.
By the end of this activity, you will discover one of the most famous theorems in all of mathematics
by looking at the relationship between the side lengths of a right triangle.
Consider the right triangle drawn to the right (figure #1). Each of the legs
has a length of 1 un. Suppose you drew the squares on the legs and on the
hypotenuse of the triangle (figure #2). Each square will have a side length
equal to a leg or equal to the hypotenuse.
What is the area of the square on leg #1?
What is the area of the square on leg #2?
What is the area of the square on the hypotenuse?
You will continue to create squares that have the same side lengths as the two legs and the
hypotenuse. Then you will find the areas of the squares you have drawn. Finally, you will
record your answers in the table provided on page number three.
Given the triangles found on page three, here are your directions:
1) Find the length of each of the legs. Record your answer in the table provided on page
two.
2) Draw in the squares of the legs and the hypotenuse. (Some have been done for you to get
you started.)
3) Find the areas of the squares of the legs and of the squares of hypotenuse. Record your
answers in table provide on page two.
4) Find the length of the hypotenuse in square root form and rounded to the nearest 100th.
Adapted from Lappon, Fey et al. Looking For Pythagoras. (Connected Mathematics), Dale
Seymour Publications © 1998
Adapted from Lappon, Fey et al. Looking For Pythagoras. (Connected Mathematics), Dale
Seymour Publications © 1998
Triangle
Length
of Leg
One
Length
of Leg
Two
Area of
Square
w/ Leg
One
Area of
Square
w/ Leg
Two
#1
1
1
1
1
a
b
Length of
Area of
Hypotenuse
square w/
(leave as a
Hypotenuse
square root)
2
2
Length of
Hypotenuse
(round to the
nearest 100th)
1.41
#2
#3
#4
#5
#6
#7
#8
#9
c
Questions:
1. Look for a pattern in the areas of the three squares (from both legs and from the hypotenuse)
of each triangle (the shaded area in the table). Use this pattern to make a hypothesis about
the relationship between the three sides of a triangle.
2. Imagine that there is a new triangle, triangle #9, that has legs that have lengths “a” and “b”
and a hypotenuse of length “c”. Please fill in the missing information in the above table.
3. If a right triangle has legs of length 5cm and 8cm, what is the length of the hypotenuse? If
necessary, draw the triangle on dot paper.
Adapted from Lappon, Fey et al. Looking For Pythagoras. (Connected Mathematics), Dale
Seymour Publications © 1998
Adapted from Lappon, Fey et al. Looking For Pythagoras. (Connected Mathematics), Dale
Seymour Publications © 1998