91Ó°ÊÓ

The midpoint formula is a straightforward way to find the exact middle point between two points on a Cartesian plane. Imagine you have two points, let's call them

• Point A: \( x_1, y_1 \)
• Point B: \( x_2, y_2 \)

The midpoint, denoted as \( M \), is found using the formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
This formula simply averages the x-coordinates and the y-coordinates of the given points, providing a new point \( (Mx, My) \) that lies exactly halfway between them.
In our example with points A(-4, -3) and B(6, 1), we applied the midpoint formula to find the midpoint: \( M = \left( 1, -1 \right) \).
This midpoint is crucial because the perpendicular bisector, which is the line we are interested in, must pass through this point.

The slope of a line measures how steep the line is, essentially showing the rate of change in y for each unit change in x.
For a line segment connecting two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the slope \( m \) is calculated as: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Example: For points A(-4, -3) and B(6, 1), the slope is calculated as \( \frac{4}{10} \), which simplifies to \( \frac{2}{5} \).
This means for every 5 units moved horizontally, the line rises 2 units.

• If the slope is positive, as in this case, the line rises to the right.
• If the slope is zero, the line is horizontal.
• If the slope is negative, the line falls to the right.

The point-slope form is a way to write the equation of a line when we know a point on the line and the slope. It's very useful, especially when working with perpendicular bisectors or parallel lines.
The general formula is: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a known point on the line and \( m \) is the slope.
Applying this form, you can conveniently express the line's equation. In our scenario, with midpoint \( (1, -1) \) and slope \( -\frac{5}{2} \), the point-slope form becomes: \( y + 1 = -\frac{5}{2}(x - 1) \).
After simplifying, we reach the equation: \( y = -\frac{5}{2}x + \frac{3}{2} \). This clean equation effectively represents the perpendicular bisector of segment AB.

A negative reciprocal is a concept often used in geometry, especially when dealing with perpendicular lines.
The negative reciprocal of a slope is the value needed to make two lines perpendicular. If one line has a slope \( m \), a perpendicular line will have a slope of \( -\frac{1}{m} \).
Example: For a slope \( \frac{2}{5} \), the negative reciprocal is \( -\frac{5}{2} \).
This is crucial because perpendicular bisectors always have negative reciprocal slopes compared with the original line segment.

• This ensures that the lines intersect at right angles.
• The product of the slopes of two perpendicular lines is always -1, which is a quick check for accuracy.
• Understanding this property helps in solving many geometric problems involving perpendicular lines.