MATEMATIKA Kelas 8 - Teorema Phytagoras | GIA Academy

Introduction

In this tutorial, we will explore the Pythagorean Theorem, a fundamental concept in mathematics, particularly in geometry. This theorem helps us understand the relationship between the sides of a right triangle. By the end of this guide, you will be able to apply the Pythagorean Theorem to solve problems involving right triangles.

Step 1: Understanding the Pythagorean Theorem

The Pythagorean Theorem states that in a right 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. This can be expressed with the formula:

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

Where:

• ( c ) is the length of the hypotenuse
• ( a ) and ( b ) are the lengths of the other two sides

Practical Tip

• Always identify the hypotenuse in a right triangle. It is the longest side and is opposite the right angle.

Step 2: Applying the Theorem

To apply the Pythagorean Theorem, follow these steps:

1. Identify the Triangle: Confirm that the triangle you are working with is a right triangle.
2. Label the Sides: Assign labels to the sides of the triangle. Typically, the hypotenuse is labeled as ( c ), while the other two sides are ( a ) and ( b ).
3. Substitute Values: Plug the known values of ( a ) and ( b ) into the formula.
4. Calculate: Solve for ( c ) or for one of the other sides if ( c ) is known.

Example

If ( a = 3 ) units and ( b = 4 ) units:

• Plug the values into the formula:

  [ c^2 = 3^2 + 4^2 ] [ c^2 = 9 + 16 ] [ c^2 = 25 ]

• Take the square root of both sides:

  [ c = 5 \text{ units} ]

Common Pitfall

• Ensure you are working with a right triangle. If the triangle is not a right triangle, the Pythagorean Theorem cannot be applied.

Step 3: Solving for a Side

If you know the hypotenuse and one side, you can rearrange the formula to find the unknown side:

1. For finding ( a ): [ a^2 = c^2 - b^2 ]

2. For finding ( b ): [ b^2 = c^2 - a^2 ]

Example

If ( c = 10 ) units and ( b = 6 ) units:

• To find ( a ):

  [ a^2 = 10^2 - 6^2 ] [ a^2 = 100 - 36 ] [ a^2 = 64 ]

• Thus, ( a = 8 ) units.

Step 4: Real-World Applications

The Pythagorean Theorem is not just theoretical; it has practical applications including:

• Construction: Ensuring structures are built correctly and safely.
• Navigation: Calculating distances on maps.
• Design: Creating accurate layouts in graphic design and architecture.

Conclusion

The Pythagorean Theorem is a powerful tool in mathematics that allows us to solve for unknown lengths in right triangles. Understanding and applying this theorem can aid in various real-world scenarios. Practice using the theorem with different triangle dimensions to enhance your skills. The more you apply it, the more comfortable you will become with its concepts and calculations.