Pythagorean theorem

(noun)

A fundamental relation among the sides of a right triangle, which states that $a^2 + b^2 = c^2$ where $c$ is the hypotenuse, and $a$ and $b$ are the lengths of the other two sides.

Related Terms

(noun)

The sum of the areas of the two squares on the legs ($a$ and $b$) is equal to the area of the square on the hypotenuse ($c$).  The formula is $a^2+b^2=c^2$.

Related Terms

Examples of Pythagorean theorem in the following topics:

  • Right Triangles and the Pythagorean Theorem

    • The Pythagorean Theorem, ${\displaystyle a^{2}+b^{2}=c^{2},}$ can be used to find the length of any side of a right triangle.
    • The Pythagorean Theorem, also known as Pythagoras' Theorem, is a fundamental relation in Euclidean geometry.
    • The theorem can be written as an equation relating the lengths of the sides $a$, $b$ and $c$, often called the "Pythagorean equation":[1]
    • The Pythagorean Theorem can be used to find the value of a missing side length in a right triangle.
    • Use the Pythagorean Theorem to find the length of a side of a right triangle

  • The Distance Formula and Midpoints of Segments

    • This formula is easily derived by constructing a right triangle with the hypotenuse connecting the two points ($c$) and two legs drawn from the each of the two points to intersect each other ($a$ and $b$), (see image below) and applying the Pythagorean theorem.
    • This theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides .
    • Substitute the values into the distance formula that is derived from the Pythagorean Theorem:
    • The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

  • Scientific Applications of Quadratic Functions

    • The Pythagorean Theorem is used to relate the three sides of right triangles.
    • In practice, the Pythagorean Theorem is often solved via factoring or completing the square.  

  • Converting Between Polar and Cartesian Coordinates

    • Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.

  • Pythagorean Identities

    • The Pythagorean identities are useful in simplifying expressions with trigonometric functions.
    • Additional identities can be derived from the Pythagorean identity $\cos^2 t + \sin^2 t = 1$.
    • We can now substitute $1$ for $\sin^2 t + \cos^2 t$, applying the Pythagorean identity:
    • Recall that one of the Pythagorean identities states $1 + \tan^2 t = \sec^2 t$.
    • Connect the trigonometric functions to the Pythagorean Theorem in order to derive the Pythagorean identities

  • The Law of Cosines

    • The Law of Cosines is a more general form of the Pythagorean theorem, which holds only for right triangles.
    • Thus, for right triangles, the Law of Cosines reduces to the Pythagorean theorem:

  • Complex Conjugates

    • This comes from the Pythagorean Theorem.

  • Trigonometry and Complex Numbers: De Moivre's Theorem

  • The Intermediate Value Theorem

    • The theorem depends on (and is actually equivalent to) the completeness of the real numbers.
    • In plotting a continuous and smooth function between two points, all points on the function between the extremes are described and predicted by the Intermediate Value Theorem.
    • It meets the requirements of the Intermediate Value Theorem.
    • In what situation would it not meet the requirements for the theorem?
    • Explain what the intermediate value theorem means for the graphs of polynomials

  • The Fundamental Theorem of Algebra

    • The fundamental theorem states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
    • The fundamental theorem of algebra says that every non-constant polynomial in a single variable $z$, so any polynomial of the form
    • There are lots of proofs of the fundamental theorem of algebra.
    • For a general polynomial $f(x)$ of degree $n$, the fundamental theorem of algebra says that we can find one root $x_0$ of $f(x)$.
    • So an alternative statement of the fundamental theorem of algebra is: