1. To understand how the Distance Formula ensures that the distance between two different points is positive, let's first review the formula itself.
The Distance Formula is expressed as:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Where (x1, y1) and (x2, y2) are the coordinates of the two points in consideration, and d represents the distance between them.
The formula calculates the Euclidean distance, which is the straight-line distance between the two points in a Cartesian coordinate system.
Now, let's consider what each part of the formula represents. The terms (x2 - x1) and (y2 - y1) represent the differences between the x-coordinates and y-coordinates of the points, respectively. These differences can be seen as the lengths of the horizontal and vertical sides, respectively, of a right-angled triangle formed by the two points.
By squaring both of these differences ((x2 - x1)^2 and (y2 - y1)^2), we ensure that they become positive numbers, as squaring any real number will result in a positive value. This step is necessary to preserve the magnitudes of the differences, regardless of their signs.
Adding up the squared differences gives us the sum of the squares of the lengths of the sides of the right-angled triangle. Then taking the square root of this sum (sqrt(...)) calculates the length of the hypotenuse, which corresponds to the distance between the two points.
Since square roots are only defined for non-negative numbers, the output of the Distance Formula will always be a non-negative value. In the case where the two points are different, the length of the hypotenuse is always positive, ensuring that the distance between them is also positive.
2. The formula for finding the midpoint on a number line cannot be directly used to find the midpoint of two points on a coordinate plane. Let's explore why this is the case.
On a number line, the midpoint formula is simply:
m = (a + b) / 2
where 'a' and 'b' represent the coordinates of the two points, and 'm' represents the midpoint between them.
This formula works because on a number line, there is only one dimension (a line), and no additional axes. Therefore, we only need to consider the absolute values of the coordinates and take their average.
However, on a coordinate plane, we have two dimensions: the x-axis and the y-axis. To find the midpoint between two points, we need to consider both x and y coordinates simultaneously.
The formula for the midpoint on a coordinate plane is:
(m_x, m_y) = ((x1 + x2)/2, (y1 + y2)/2)
This formula accounts for the differences in both the x and y coordinates of the two points and takes their average separately for each axis.
By using this formula, we ensure that the midpoint falls exactly halfway along both the x and y axes, the average of their respective coordinates. This is not achievable with the formula for calculating the midpoint on a number line, as that formula only considers one dimension and would not account for the y-coordinate of the points on the coordinate plane.