Analisis Bangun Ruang (Dimensi Tiga) Matematika Wajib Kelas XII Bagian 1 - Jarak Titik ke Titik

Introduction

This tutorial provides a step-by-step guide on calculating the distance between two points in three-dimensional space, based on the mathematics curriculum for 12th-grade students. Understanding how to calculate this distance is fundamental in geometry and can be applied in various fields, including engineering, physics, and computer graphics.

Step 1: Understanding the Distance Formula in 3D

To find the distance between two points in three-dimensional space, you need to use the distance formula. For points A(x1, y1, z1) and B(x2, y2, z2), the formula is:

Distance = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²)

Practical Advice

• Ensure you know the coordinates of both points before applying the formula.
• Pay attention to signs; negative coordinates are valid and must be included in calculations.

Step 2: Example Calculation

Let’s go through an example to clarify how to use the formula.

Example

Consider points A(2, 3, 5) and B(6, 7, 9).

1. Identify the coordinates:

   • A: (2, 3, 5)
   • B: (6, 7, 9)

2. Substitute the coordinates into the distance formula:

   Distance = √((6 - 2)² + (7 - 3)² + (9 - 5)²)
   

3. Calculate the differences:

   • (6 - 2) = 4
   • (7 - 3) = 4
   • (9 - 5) = 4

4. Plug the differences back into the formula:

   Distance = √(4² + 4² + 4²)
   Distance = √(16 + 16 + 16)
   Distance = √48
   

5. Simplify:

   Distance = 4√3
   

Practical Advice

• When simplifying square roots, look for perfect squares to make calculations easier.
• Double-check each step to avoid arithmetic errors.

Step 3: Applying the Distance Formula

Once you are comfortable with the formula and its calculations, practice using it with different pairs of points to strengthen your understanding.

Common Pitfalls to Avoid

• Forgetting to square the differences.
• Confusing the order of coordinates (always maintain the order: x1, y1, z1 and x2, y2, z2).
• Neglecting to simplify the final answer, which can often lead to more complicated forms.

Conclusion

In this tutorial, you learned how to calculate the distance between two points in three-dimensional space using the distance formula. Practice with various examples to enhance your skills. For further study, explore the concepts of distance from a point to a line and a plane, which will be covered in subsequent tutorials.