Pythagorean Theorem

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.  

• 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

• 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 $z$, 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: