To determine the formula used to find the midpoint between two coordinate points, let's examine each option provided and understand what it describes.
- Option a: [tex]\( M = \frac{x_1 + x_2}{2} \)[/tex]
This formula only calculates the average of the x-coordinates of two points. It does not include the y-coordinates, and hence, it does not fully describe the midpoint of two points in a plane.
- Option b: [tex]\( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)[/tex]
This formula calculates the average of the x-coordinates and y-coordinates separately. The result is expressed as a coordinate pair, which correctly represents the midpoint of two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]. This is indeed the formula used to find the midpoint between two coordinate points.
- Option c: [tex]\( m = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex]
This formula represents the slope of the line passing through the two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]. It is used to find the steepness and direction of the line, not the midpoint.
- Option d: [tex]\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]
This formula is used to calculate the Euclidean distance between two points in a plane. It tells us how far apart the two points are, but it does not give us the midpoint.
Among the given options, it is clear that:
Option b: [tex]\( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)[/tex]
is the correct formula to find the midpoint between two coordinate points.
