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

• Arc Length and Speed

  • Arc length and speed in parametric equations can be calculated using integration and the Pythagorean theorem.
  • This equation is obtained using the Pythagorean Theorem.
  • By the Pythagorean Theorem, each hypotenuse will have length $\sqrt{dx^2 + dy^2}$.
  • As shown previously using the Pythagorean Theorem, it is given by:

• 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.  

• Arc Length and Speed

  • To find this using integration, we should start out by using the Pythagorean Theorem for length of the different sides of a triangle:
  • For a small piece of curve, ∆s can be approximated with the Pythagorean theorem.

• 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

• Superposition of Fields

  • An overall resultant vector can be found by using the Pythagorean theorem to find the resultant (the hypotenuse of the triangle created with applied forces as legs) and the angle with respect to a given axis by equating the inverse tangent of the angle to the ratio of the forces of the adjacent and opposite legs.

• 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:

• Arc Length and Surface Area

  • According to Pythagoras's theorem $ds^2=dx^2+dy^2$, from which:
  • For a small piece of curve, $\Delta s$ can be approximated with the Pythagorean theorem.