The Pythagorean Theorem

One of the foremost fundamental results is that the well-known Pythagorean Theorem.

This states that a a2 + b2 = c2 in a right triangle with sides a and b and hypotenuse c.

The figure to the proper indicates one among the various known proofs of this fundamental result. Indeed, the area of the “big” square is (a + b)2 and can be decomposed into the area of the smaller square plus the areas of the four congruent triangles.

Pythagorean Theorem

That is,

(a + b)2 = c2 + 2ab

which immediately reduces to a2 + b2 = c2


sum of the interior angles

Next, we recall the equally well-known result that the sum of the interior angles of a triangle is 180 . The proof is easily inferred from the diagram to the right.


Exercises:

01.

Pythagorean Theorem exercise 1

Prove Euclid’s Theorem for Proportional Segments, i.e., given the right triangle ΔABC as indicated, then
h2 = pq, a2 = pc, b2 = qc

02. Prove that the sum of the interior angles of a quadrilateral ABCD is 360.

03.

excercise 2

In the diagram to the right, ΔABC is a right triangle, segments [AB] and [AF] are perpendicular and equal in length, and [EF] is perpendicular to [CE]. Set a = BC, b = AB, c = AB, and deduce President Garfield’s proof* of the Pythagorean theorem by computing the area of the trapezoid BCEF.


*James Abram Garfield (1831–1881) published this proof in 1876 in the Journal of Education (Volume 3 Issue 161). While a member of the House of Representatives. He was assassinated in 1881 by Charles Julius Guiteau. As an aside, notice that Garfield’s diagram also provides a simple proof of the fact that perpendicular lines in the planes have slopes which are negative reciprocals.

Related lessons: Trigonometric Identities

FAQ about Pythagorean Theorem

What is the formula of Pythagoras?

The theorem, also referred to as the Pythagorean theorem, states that the square of the length of the hypotenuse is adequate to the sum of squares of the lengths of the opposite two sides of the right triangle.
Hypotenuse2  =  Perpendicular2 + Base2 

What is meant by Pythagoras Theorem?

The Pythagoras theorem also referred to as Pythagorean theorem is employed to seek out the sides of a right angled triangle. This theorem is usually utilized in Trigonometry, where we use trigonometric ratios like sine, cos, tan to seek out the length of the edges of the right triangle

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