Introduction

The concept of the perpendicular bisector is fundamental in geometry. It is the line segment that not only bisects a given line segment at a 90-degree angle but also passes through its midpoint. This article will explore a distinct and enjoyable method to derive the equation of the perpendicular bisector through two given points, 51 and -37. We will delve into the rigorous process of calculating the slope, determining the midpoint, and finally, formulating the equation of the line using a different approach compared to the typical methods.

Calculating the Slope

To begin, we need to determine the slope of the line connecting the two points, 51 and -37. Let's denote the coordinates of the points as (x1, y1) (51, 2) and (x2, y2) (-37, 7).

Step 1: Calculate the Slope

The formula for the slope (m) between two points (x1, y1) and (x2, y2) is given by:

m (y2 - y1) / (x2 - x1)

m (7 - 2) / (-37 - 51) 5 / -88 -5/88

Step 2: Find the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the given slope. Therefore, the slope (m') of the perpendicular bisector is:

m' -1 / (-5/88) 88/5

This slope represents the rate at which the perpendicular line rises relative to the run.

Finding the Midpoint

The next step is to identify the midpoint of the line segment joining the points (51, 2) and (-37, 7). The midpoint formula is:

Midpoint ((x1 x2)/2, (y1 y2)/2)

Substituting the coordinates of the points:

Midpoint ((51 - 37)/2, (2 7)/2) (14/2, 9/2) (7, 4.5)

The midpoint of the line segment is (7, 4.5).

Deriving the Equation of the Perpendicular Bisector

To derive the equation of the perpendicular bisector, we use the point-slope form of the line equation, which is:

y - y1 m'(x - x1)

Using the slope (88/5) and the midpoint (7, 4.5):

y - 4.5 88/5(x - 7)

After simplifying this, we get:

y - 4.5 88/5x - 1232/5

y 88/5x - (1232/5 - 22.5)

y 88/5x - 1185/5

y 88/5x - 237

Hence, the equation of the line that is the perpendicular bisector of the line segment connecting the points (51, 2) and (-37, 7) is:

y 88/5x - 237

This is a unique and specific equation, as illustrated in the Desmos graph here.

Conclusion

The method to find the equation of the perpendicular bisector is not only an essential aspect of geometry but also a fascinating way to engage with mathematical concepts. By following a systematic and engaging approach, you can appreciate the intricacies of this unique equation. This article presents a distinct and enjoyable method to calculate the slope, determine the midpoint, and finally, formulate the equation, providing a different perspective to teachers and students alike.

About the Author

The author is a seasoned SEO expert at Google. With extensive experience in optimizing content for search engines, they have successfully applied advanced SEO techniques to create unique and informative articles. Their expertise in combining mathematical principles with SEO practices makes them an authority in this domain.

Desmos Graph

For a visual representation, explore the Desmos graph of the following equation:

x - 52y - 12 x32y - 72