Pythagorean Theorem and Triangles

Right Triangles

• A right triangle is a triangle with one right angle (90 degrees)
• In a right triangle, the Pythagorean theorem can be applied to find the length of the hypotenuse
• The theorem only works for right triangles

Distance Formula

• The distance formula is a direct application of the Pythagorean theorem
• It is used to find the distance between two points in a coordinate plane
• Formula: √((x2 - x1)^2 + (y2 - y1)^2)

Similar Triangles

• Similar triangles are triangles with the same shape but not necessarily the same size
• If two triangles are similar, their corresponding sides are proportional
• The Pythagorean theorem can be used to prove the similarity of triangles

Hypotenuse

• The hypotenuse is the longest side of a right triangle, opposite the right angle
• It is the side opposite the 90-degree angle
• The Pythagorean theorem is used to find the length of the hypotenuse

Proofs

• There are many different proofs of the Pythagorean theorem
• Some common proofs include:
  • Geometric proof: using the area of squares on each side of the triangle
  • Algebraic proof: using the equation a^2 + b^2 = c^2
  • Similar triangles proof: using the concept of similar triangles

Applications

• The Pythagorean theorem has many real-world applications, including:
  • Building design and architecture
  • Physics and engineering
  • Computer graphics and game development
  • Trigonometry and navigation
  • Land surveying and mapping