Pythagoras (1) - Pengenalan Teorema Pythagoras, Pythagoras Theorem - Matematika SMP

Introduction

This tutorial introduces the Pythagorean Theorem, a fundamental concept in mathematics that is crucial for students, particularly in middle school. Understanding this theorem allows you to solve problems involving right-angled triangles and has practical applications in various fields, such as construction and navigation.

Step 1: Understanding the Pythagorean Theorem

• The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

• Mathematically, it is expressed as:

  a² + b² = c²
  

  where:

  • a and b are the lengths of the two legs,
  • c is the length of the hypotenuse.

Step 2: Proof of the Pythagorean Theorem

• The theorem can be proven using various methods, including geometric rearrangement and algebraic manipulation.
• A common visual proof involves creating a square on each side of the triangle and showing that the area of the squares on the legs combined equals the area of the square on the hypotenuse.

Step 3: Solving Basic Pythagorean Problems

• To apply the theorem, identify the lengths of two sides of a right triangle and solve for the third.

• Example Problem:

  • Given a triangle with legs of lengths 3 and 4, find the hypotenuse.

  c² = 3² + 4²
  c² = 9 + 16
  c² = 25
  c = √25
  c = 5
  

Step 4: Pythagorean Theorem for Obtuse Triangles

• Understand that the Pythagorean Theorem applies strictly to right triangles. However, it can be adapted for obtuse triangles using the concept of the extended version of the theorem.

• For obtuse triangles, the relation is:

  c² > a² + b²
  

Step 5: Pythagorean Theorem in Overlapping Triangles

• When working with overlapping triangles, apply the theorem to each segment.
• Use the sum of the areas or lengths from the triangles to find unknown values.

Step 6: Practical Applications of the Pythagorean Theorem

• Ladders: Determine the length of a ladder needed to reach a certain height when placed at an angle.

  • Example: If a ladder needs to reach 12 feet high and is placed 5 feet away from the wall, find the ladder length (hypotenuse).

  c² = 12² + 5²
  c² = 144 + 25
  c² = 169
  c = √169
  c = 13 feet
  

• Buildings: Calculate the height of a building using the distance from the base and the angle of elevation.

Conclusion

The Pythagorean Theorem is a powerful tool in geometry that provides solutions for various real-world problems involving right triangles. By understanding its principles, proofs, and applications, you can effectively tackle mathematical challenges in school and beyond. For further practice, try solving different problems involving right triangles and explore more complex applications in construction and navigation.