Pythagorean Theorem

Pythagorean Theorem in Algebra

by

Pythagorean Theorem in Algebra

Over 2,500 years ago, a Greek mathematician named Pythagoras popularized the concept that a relationship exists between the hypotenuse and the legs of right triangles and that this relationship is true for all right triangles. The Egyptians knew of this concept, as it related to 3, 4, 5 right triangles, long before the time of Pythagoras. It was Pythagoras, however, who generalized the concept and who is attributed with its first geometrical demonstration. Thus, it has become known as the Pythagorean Theorem.

“In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.”
pythagorean-theorem

Pythagorean Triples:
There are specific sets of positive integer values that have a special connection to the Pythagorean Theorem. These sets not only satisfy the Pythagorean Theorem, but multiples of these integers also satisfy the Pythagorean Theorem.

For example, the numbers 3, 4, 5 satisfy the Pythagorean Theorem: 32 + 42 = 52.
If you multiply all of these numbers by 2, you get 6, 8, 10 which also statisfy the Pythagorean Theorem: 62 + 82 = 102. Multiplying 3, 4, 5 by the same positive integer value will create a new set of integers that will also satisfy the Pythagorean Theorem.
Pythagorean Theorem in Algebra 2
If a, b and c are positive integers that satisfy the Pythagorean Theorem, then ka, kb and kc, where k is a positive integer, will also satisfy the Pythagorean Theorem.
The special sets of integers that possess this property are called “Pythagorean Triples”.

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.
Pythagorean Theorem in Algebra 1
The theorem can be rephrased as, “The (area of the) square described upon the hypotenuse of a right triangle is equl to the sum of the (areas of the) squares described upon the other two sides.”

REMEMBER: The Pythagorean Theorem ONLY works in Right Triangles!

Example 1: A triangle has sides 6, 7 and 10. Is it a right triangle?
Let a = 6, b = 7 and c = 10. The longest side MUST be the hypotenuse, so c = 10. Now, check to see if the Pythagorean Theorem is true.
102 = 62 + 72
100 = 36 + 49
100 = 85
Since the Pythagorean Theorem is NOT true, this triangle is NOT a right triangle.

Example 2: A ramp was constructed to load a truck. If the ramp is 9 feet long and the horizontal distance from the bottom of the ramp to the truck is 7 feet, what is the vertical height of the ramp to the nearest tenth of a foot?
pythagorean-theorem-5Since the ramp is described as having horizontal and vertical measurements, a right angle is implied. Solve using the Pythagorean Theorem:
92 = x2 + 72
81 = x2 + 49
x2 = 81 -49
x2 = 32
x = 5.7
The height of the ramp is 5.7 feet. The ramp will allow packages to be loaded into an area of the truck that is too high to be reached from the ground.

Pythagorean Theorem

by

Pythagorean Theorem

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.
(c2 = a2 + b2)
It is believed that this theorem was known, in some form, long before the time of Pythagoras. A thousand years before Pythagoras, the Babylonians recorded their knowledge of the theorem on clay tablets. The Egyptians used the concept of the theorem in the building of the pyramids. In 1100 B.C. China, Tschou-Gun knew of this theorem.
Pythagorean Theorem 1It was, however, Pythagoras who generalized the theorem to all right triangles and is credited with its first geometrical demonstration. Consequently, the theorem bears his name.

Geometrical Proof: The Pythagorean Theorem has drawn a good deal of attention from mathematicians. There are hundreds of geometrical proofs (or demonstrations) of the theorem, with even a larger number of algebraic proofs.
Pythagorean Theorem 2Geometrically, the Pythagorean Theorem can be interpreted as discussing the areas of squares whose sides are the sides of the triangle (as seen in the picture at the left). 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.”

Converse:

Theorem: If a triangle is 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.
Converse: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, the triangle is a right triangle.

Pythagorean Triples: There are certain sets of numbers that have a very special property in connection to the Pythagorean Theorem. Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem.

For example: the numbers 3, 4, and 5 satisfy the Pythagorean Theorem. If you multiply all three numbers by 2 (you will get 6, 8, and 10), these new numbers ALSO satisfy the Pythagorean theorem.

The special sets of numbers that possess this property are called Pythagorean Triples.
The most common Pythagorean Triples are:
Pythagorean Theorem 3

There are equations for generating Pythagorean Triples. Such equationss are used by computer programmers to generate lists of Pythagorean Triples. Equations: When m and n are both positive integers and m < n, Pythagorean Triples can be generated using the following equations where a and b will be the legs of the right triangle and c will be the hypotenuse.
Pythagorean Theorem 4It can be shown that the results from these equations do, in fact, satisfy the Pythatorean Theorem:
Pythagorean Theorem 5

Connection to the Distance Formula: The Pythagorean Theorem serves as the basis for determining the distance between two points in the Cartesian plane, called the Distance Formula.
If (x1, y1) and (x2, y2) are points in the plane, then the distance between them can be determined using the Pythagorean Theorem.
Pythagorean Theorem 6Pythagorean Theorem 7

Generalization of the Law of Cosines: The Pythagorean theorem is a specific case of the more general theorem relating the lengths of sides in any triangle, called the Law of Cosines.
Law of Cosines: c2 = a2 + b2 – 2ab cosC
When angle C is 90°, cosC equals 0 and the formula becomes to the Pythagorean Theorem.

In non-Euclidean Geometries: While the Pythagorean Theorem is of prime importance to Euclidean Geometry (and is derived from its axioms), it does not hold in non-Euclidean Geometries.