Pythagorean Theorem

(noun)

States that the square of the hypotenuse is equal to the sum of the squares of the other two sides in a right triangle.

Related Terms

  • midpoint
  • distance

Examples of Pythagorean Theorem in the following topics:

  • Right Triangles and the Pythagorean Theorem

    • The Pythagorean Theorem, a2+b2=c2,{\displaystyle a^{2}+b^{2}=c^{2},}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 aaa, bbb and ccc, 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 (ccc) and two legs drawn from the each of the two points to intersect each other (aaa and bbb), (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.  
  • Pythagorean Identities

    • The Pythagorean identities are useful in simplifying expressions with trigonometric functions.
    • Additional identities can be derived from the Pythagorean identity cos2t+sin2t=1\cos^2 t + \sin^2 t = 1cos​2​​t+sin​2​​t=1.
    • We can now substitute 111 for sin2t+cos2t\sin^2 t + \cos^2 tsin​2​​t+cos​2​​t, applying the Pythagorean identity:
    • Recall that one of the Pythagorean identities states 1+tan2t=sec2t1 + \tan^2 t = \sec^2 t1+tan​2​​t=sec​2​​t.
    • Connect the trigonometric functions to the Pythagorean Theorem in order to derive the Pythagorean identities
  • Converting Between Polar and Cartesian Coordinates

    • Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.
  • 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 zzz, so any polynomial of the form
    • There are lots of proofs of the fundamental theorem of algebra.
    • For a general polynomial f(x)f(x)f(x) of degree nnn, the fundamental theorem of algebra says that we can find one root x0x_0x​0​​ of f(x)f(x)f(x).
    • So an alternative statement of the fundamental theorem of algebra is:
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