Pythagorean Theorem

The Pythagorean Theorem, a cornerstone of geometry, relates the sides of a right triangle. At MathMultiverse, we explore its applications, from simple triangles to 3D space, with interactive visualizations and clear examples.

Formula

For a right triangle with legs \( a \), \( b \), and hypotenuse \( c \):

\[ a^2 + b^2 = c^2 \]

Rearrangements: \( c = \sqrt{a^2 + b^2} \), \( a = \sqrt{c^2 - b^2} \), \( b = \sqrt{c^2 - a^2} \).

Examples

3-4-5 Triangle

Legs \( a = 3 \), \( b = 4 \):

\[ c = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \]

Decimal Sides

Legs \( a = 2.5 \), \( b = 6.2 \):

\[ c = \sqrt{2.5^2 + 6.2^2} \approx 6.69 \]

3D Distance

Points \( (0, 0, 0) \), \( (3, 4, 5) \):

\[ d = \sqrt{3^2 + 4^2 + 5^2} \approx 7.07 \]

Ladder Problem

Ladder 10 m, base 6 m:

\[ b = \sqrt{10^2 - 6^2} = 8 \]

Non-Right Triangle

Sides 5, 6, 7:

\[ 5^2 + 6^2 = 61 \neq 49 = 7^2 \]

Pythagorean Triple

Scale 5-12-13 by 2:

\[ 10^2 + 24^2 = 26^2 \]

Coordinate Geometry

Points \( (1, 2) \), \( (4, 6) \):

\[ d = \sqrt{(4-1)^2 + (6-2)^2} = 5 \]

Proofs

Area Proof

Square with side \( a + b \), four triangles, inner square \( c \):

\[ (a + b)^2 = 4 \cdot \frac{1}{2}ab + c^2 \] \[ a^2 + b^2 = c^2 \]

Similarity Proof

Altitude divides triangle:

\[ \frac{a}{c} = \frac{h}{b}, \quad ab = ch \] \[ a^2 + b^2 = c^2 \]

Visualizations

Right Triangle

Applications

  • Room Diagonal: \( \sqrt{5^2 + 4^2} \approx 6.40 \, \text{m} \).
  • Navigation: \( \sqrt{8^2 + 6^2} = 10 \, \text{km} \).
  • Construction: Verify 6-8-10 triangle.
  • Box Diagonal: \( \sqrt{3^2 + 4^2 + 5^2} \approx 7.07 \, \text{m} \).
  • Surveying: \( \sqrt{15^2 + 20^2} = 25 \, \text{m} \).
  • Coordinate Geometry: Distance \( (-2, 3) \) to \( (4, -1) \approx 7.21 \).