Pythagorean Theorem
c2 = a2 + b2

When labeling the diagram, a small letter "a" is the length of the side opposite the angle whose vertex is labeled with capital "A". All sides are labeled in this same manner.
label

dividerdash

Interpretation:

The Pythagorean Theorem can be interpreted in relation to squares drawn to coincide with each of the sides of a right triangle, as shown at the right.

The theorem can be rephrased as, "The (area of the) square described upon the hypotenuse of a right triangle is equal to the sum of the (areas of the) squares described upon the other two sides."

Remember that the area of a square with a side length of "a" is a × a or a2.

pythsquares

dividerdash

There are numerous methods for verifying that the Pythagorean Theorem is true.
We will be examining just a few of these methods.

Verify the Pythagorean Theorem:


bullet Square in a Square Approach:
The blue right triangle, as shown, is copied and arranged in a manner that forms a large square (using the legs) and an inner square (using the hypotenuses).

The four blue triangles are congruent. They each have a right angle and legs of length a and b. (SAS)

We can verify that the inner square is actually a square. We know that in a right triangle, the acute angles add to 90º.    m∠1 + m∠2 = 90.
Now, m∠1 + m∠2 + m∠3 = 180 since the three angles for a straight line. Thus m∠3 must be 90º, verifying that the inner quadrilateral is actually a square with 4 right angles.


squaresquare
Congruent triangles have the same area.
The length of a side of the large square is a + b.
The area of the large square is (a+b) • (a+b).
The area of the inner square is c • c.
The area of a blue triangle is ½ab.

Area of large square = area of inner square + area of the 4 blue right triangles.
(a+b)(a+b) = c • c + 4(½ab)
a2 + 2ab + b2 = c2 + 2ab (now, subtract 2ab from both sides)
a2 + b2 = c2

dividerdash

bullet Rearranging the Diagram:
If we re-draw the sub-divisions of the square seen in the example above, we can see the Pythagorean Theorem pop out.

Both squares shown below are the same size, with a side length of a+b. They are each of the same area size.
The eight blue triangles in the diagrams are congruent. The are right triangles with legs of equal lengths. So, all blue triangles have the same area.
squarerectangle
squaresquare

If we remove the four congruent blue triangles from each drawing, the remaining portions will represent equal area. Representing the remaining areas, we see the Pythagorean Theorem.

SquarerectangleMT
squaresquareMT
a2 + b2 = c2

dividerdash


In high school Geometry, you will see that there are both geometrical and algebraic "proofs" of the Pythagorean Theorem. One such geometric proof is shown below.

PROOF:

This is a geometrical proofs of the Pythagorean Theorem similar triangles.

bullet PROOF:
"If a triangle is a right triangle, then the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs."pyproofG

pyproofpic
Statements
Reasons
1.
py1
py1a
1.
Given
2.
py2
2.
The altitude of a triangle is a segment from any vertex perpendicular to the line containing the opposite side.
3.
py3
3.
Perpendicular lines form right angles.
4.
py4
4.
A right triangle contains one right angle.
5.
py5
5.
All right angles are congruent.
6.
py6
6.
Reflexive (Identity)
7.
py7
7.
AA Theorem for Similarity
8.
py8
8.
Corresponding sides of similar triangles are in proportion.
9.
py9
9.
In a proportion, the product of the means = the product of the extremes.
10.
py10
10.
Addition
11.
py11
11.
Distributive property in reverse
12.
py12
12.
Whole quantity = sum of parts.
13.

py13

13.
Substitution
14.
py14
14.
Multiplication


 


divider

NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use".