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3.1 – The Pythagorean Theorem

Objectives

  • Derive the Pythagorean theorem a^2+b^2=c^2 from the relationship between right triangles and squares.
  • Use the Pythagorean theorem to find the values of missing sides in a right triangle.
  • Prove that triangles are right triangles using the converse of the Pythagorean theorem.
  • Define and give examples of Pythagorean triples.
  • Use the Pythagorean theorem to solve real-world problems.

 

Key Terms

  • Hypotenuse – The side opposite from the right angle in a right triangle.
    • It is the triangle’s longest side.
  • Leg – Either of the two shorter sides of a right triangle.
    • Always touches the right angle.
  • Pythagorean Theorem – The theorem that relates the side lengths of a right triangle.
    • The theorem states that the square of the hypotenuse equals the sum of the squares of the legs: a^2+b^2=c^2
  • Pythagorean Triple – A set of three whole numbers, a, b, and c, that satisfies the equation a^2+b^2=c^2.
    • If the side lengths of a triangle form a Pythagorean triple, it is a right triangle.

 

Notes

Right Triangles
  • Right Triangles are:
    • Triangles with one right angle (an angle measuring 90°).
    • The other two angles in a right triangle are always acute (less than 90°).

GeoA 3.01 - Right Triangle Parts


  • Remember: the area of a square is (side)^2

GeoA 3.01 - Squares

Pythagorean Theorem
  • Pythagorean Theorem
    • Pythagorean theorem: If a triangle is a right triangle, then the square of its longest side equals the sum of the squares of its other two sides.
    • Converse: If the square of the longest side of a triangle equals the sum of the squares of its other two sides, then the triangle is a right triangle
    • Solve for a missing side of a right triangle
      • 1. Start with the Pythagorean theorem.
      • 2. Plug in the side lengths you know.
      • 3. Solve the equation for the unknown side length

GeoA 3.01 - Pythagorean Theorem

  • Examples

GeoA 3.01 - Pythagorean Theorem ex


  • Tests for Right Triangles

GeoA 3.01 - Right Triangle Test


  • Common Pythagorean Triples
    • How to Find a Pythagorean Triple:
      • Start with a Pythagorean triple you know
      • Multiply each number in that triple by the same whole number
      • The three products form another Pythagorean triple

GeoA 3.01 - Pythagorean Triples

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