91Ó°ÊÓ

The midpoint formula is a helpful tool in geometry. It allows you to find the center point of a line segment given its endpoints. This is crucial when you need to determine features like the perpendicular bisector. The formula is:

• \[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Here:

• \( x_1, y_1 \) are the coordinates of point A
• \( x_2, y_2 \) are the coordinates of point B

To find the midpoint, add the x-coordinates of your endpoints and divide by 2. Do the same for the y-coordinates. In the exercise, we find the midpoint of points \( A(1,4) \) and \( B(7,-2) \), resulting in \( M(4,1) \).
This midpoint balances the line segment, being equidistant from both A and B.

Understanding the slope of a line is key in geometry, particularly for determining perpendicularity. A slope indicates how steep a line is. The formula for finding the slope \( m \) of a line joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula calculates the change in the y-coordinates (the rise) over the change in x-coordinates (the run).
For example, in the exercise, using points \( A(1,4) \) and \( B(7,-2) \), the slope is \( \frac{-2 - 4}{7 - 1} = -1 \). This result gives us valuable information: a slope of -1 means the line descends one unit vertically for every horizontal unit, forming a diagonal line.
For a perpendicular bisector, you need the negative reciprocal of this slope, which would be 1 for our current example.

When devising a line's equation, the point-slope form is a handy approach. It expresses a line's equation through a known point and slope. The point-slope form is stated as:

• \[ y - y_1 = m(x - x_1) \]

Here:

• \( m \) is the slope
• \((x_1, y_1)\) is a point on the line

For our exercise, we use the midpoint \((4,1)\) and the perpendicular bisector's slope \(1\) to find the equation of the bisector.
Plugging in these values, we have:

• \[ y - 1 = 1(x - 4) \]

Simplifying this results in \( y = x - 3 \), giving us the perpendicular bisector's linear equation.
This procedure helps in situations where line characteristics (a particular point and slope) enable easier calculations than using other formulas.