TR-11: Distance Between Points in a Plane (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial guides you through determining the distance between two points in a plane using the Pythagorean Theorem. This concept is vital in trigonometry and has practical applications in various fields such as physics, engineering, and computer graphics. By the end of this guide, you'll understand how to apply the distance formula effectively.

Step 1: Understand the Distance Formula

To calculate the distance between two points, you can use the general distance formula derived from the Pythagorean Theorem. The formula is expressed as follows:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Where:

• (d) is the distance between the two points.
• ((x_1, y_1)) are the coordinates of the first point.
• ((x_2, y_2)) are the coordinates of the second point.

Practical Tips

• Always ensure your coordinates are correctly identified.
• Pay attention to the signs of your coordinates; subtracting can result in negative values, which will be squared in the formula.

Step 2: Using Deltas for Simplification

A simpler version of the distance formula uses 'deltas' to convey the differences in coordinates. This version is represented as:

[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} ]

Where:

• (\Delta x = x_2 - x_1)
• (\Delta y = y_2 - y_1)

Step-by-Step Calculation

1. Calculate (\Delta x) and (\Delta y):

   • Identify your two points, e.g., Point A (2, 3) and Point B (5, 7).
   • Calculate:
     • (\Delta x = 5 - 2 = 3)
     • (\Delta y = 7 - 3 = 4)

2. Substitute into the distance formula:

   • (d = \sqrt{(3)^2 + (4)^2})

3. Solve the equation:

   • (d = \sqrt{9 + 16} = \sqrt{25} = 5)

Common Pitfalls

• Forgetting to square the differences before adding them.
• Mixing up the order of points, which can lead to confusion but does not affect the distance since it is always positive.

Step 3: Real-World Applications

Understanding the distance formula can help in several real-world scenarios:

• Navigation: Determining the shortest route between two locations on a map.
• Physics: Calculating the distance traveled by an object in motion.
• Computer Graphics: Placing objects in 2D space accurately.

Conclusion

In this tutorial, you learned how to calculate the distance between two points in a plane using the distance formula derived from the Pythagorean Theorem. By practicing this formula and understanding its components, you'll be equipped to tackle problems in various fields. For further practice, consider exploring additional problems related to this topic or advance to more complex 3D space calculations.