Pythagorean theorem

Page 1: Introduction to Pythagorean Theorem

• Pythagorean Theorem Equation: c² = a² + b²

  • Where:

    • c: length of the hypotenuse

    • a: one leg of the right triangle

    • b: the other leg of the right triangle

Page 2: Understanding the Theorem

• Right Triangle Characteristics:

  • The theorem applies to right triangles only.

  • Legs: Two sides that form the right angle and are referred to as a and b.

  • Hypotenuse: The side opposite the right angle, designated as c.

• Pythagorean Theorem Relation:

  • Formula: a² + b² = c²

  • Use case: When the lengths of any two sides are known, the length of the third side can be calculated.

Page 3: Example Calculation

• To find the hypotenuse where:

  • Given: a = 10.5 cm, b = 14 cm

  • Calculation Steps:

    • a² + b² = c²

    • (10.5 cm)² + (14 cm)² = c²

    • 110.25 cm² + 196 cm² = c²

    • 306.25 cm² = c²

    • c = √306.25 cm² = 17.5 cm

  • Conclusion: The unknown side has a measure of 17.5 cm.

Page 4: Another Example

• Problem: Find the hypotenuse of a triangle where:

  • Given: a = 10 m, b = 24 m

  • Pythagorean relation: a² + b² = c²

  • Calculation of hypotenuse:

    • 10² + 24² = c²

    • 100 m² + 576 m² = c²

    • c² = 676 m²

    • c = √676 m² = 26 m

Page 5: Practice Problems

• Task: Solve for the unknown side length in the triangles:

  1. Triangle with sides 12 cm and 2.5 cm:

     • Calculation: 12² + 2.5² = x²

     • 144 cm² + 6.25 cm² = x²

     • x² = 150.25 cm²

     • x = √150.25 cm² = 12.25 cm

  2. Triangle with sides 5 cm and 2 cm:

     • Calculation: 5² + 2² = x²

     • 25 cm² + 4 cm² = x²

     • x² = 29 cm²

     • x ≈ 5.39 cm

Page 6: Multi-step Problems

• Application in Multi-step Problems:

  • Given lengths: mAB = 10 cm, mAD = 12 cm, mAC = 18 cm

  • To find mBC:

    • Using Pythagorean theorem: mAC² = mAB² + mBC²

    • 18² = 10² + mBC²

    • mBC² = 324 cm² - 100 cm² = 224 cm²

    • mBC = √224 cm ≈ 14.96 cm

  • Continue solving for other segments using provided relations and formulas.

• Calculating other values:

  • mBD, mDC, etc. can also be found through systematic application of the theorem.

  • Each side's calculation demonstrates the consistent application of c² = a² + b².