The Pythagorean Theorem
Objective Learn how to use the Pythagorean
Theorem to find the length of the side of a right triangle and to
solve problems.
The Pythagorean Theorem is probably the most famous theorem in
mathematics. Its history dates back to about 2500 years ago to
the famous Greek mathematician and philosopher Pythagoras.
However, some historians think that the theorem dates back even
further as ancient writings have been found that list
“Pythagorean triples” or triples of whole numbers that
can be the lengths of the sides of a right triangle.
Modeling the Pythagorean Theorem
Let's see a model for the Pythagorean Theorem.
The sides of the right triangle have lengths of 3, 4, and 5
units, and the areas of the squares are 9, 16, and 25 square
units, respectively. These numbers form the basis for the
Pythagorean Theorem. The area of the larger square is equal to
the total area of the two smaller squares.
5 2 = 3 2 + 4 2
25 = 9 + 16
This relationship is true for any right
triangle.
Before seeing the Pythagorean Theorem in its algebraic form,
let's check the following:
• The sides of a right triangle that are adjacent to the
right angle are called the legs of the triangle.
• The side opposite the right angle is called the hypotenuse.
The Pythagorean Theorem
In a right triangle, the square of the length of the
hypotenuse is equal to the sum of the squares of the lengths of
the legs.
c 2 = a 2 + b 2
How are the model of the Pythagorean Theorem and the
expression c 2 = a 2 + b 2
related?
Suppose the lengths of the sides of the right triangle in the
model are a, b, and c with c being the longest side. Then, the
areas of the squares would be a 2, b 2, and
c 2, respectively, and c 2 would equal the
sum of a 2 and b 2.
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