To find the coordinates of the other end of the line segment when one end and the midpoint are given, we can use the midpoint formula. The formula for the midpoint [tex]$(M_x, M_y)$[/tex] of a line segment with endpoints [tex]$(x_1, y_1)$[/tex] and [tex]$(x_2, y_2)$[/tex] is:
[tex]\[ M_x = \frac{x_1 + x_2}{2} \][/tex]
[tex]\[ M_y = \frac{y_1 + y_2}{2} \][/tex]
Given:
- The midpoint [tex]\( (M_x, M_y) = (4, 1) \)[/tex]
- One endpoint [tex]\( (x_1, y_1) = (2, 5) \)[/tex]
We need to find the coordinates of the other endpoint [tex]\( (x_2, y_2) \)[/tex].
1. First, let's use the x-coordinates:
[tex]\[ M_x = \frac{x_1 + x_2}{2} \][/tex]
Substituting the known values:
[tex]\[ 4 = \frac{2 + x_2}{2} \][/tex]
To solve for [tex]\( x_2 \)[/tex], multiply both sides by 2:
[tex]\[ 8 = 2 + x_2 \][/tex]
Then, subtract 2 from both sides:
[tex]\[ x_2 = 6 \][/tex]
2. Next, let's use the y-coordinates:
[tex]\[ M_y = \frac{y_1 + y_2}{2} \][/tex]
Substituting the known values:
[tex]\[ 1 = \frac{5 + y_2}{2} \][/tex]
To solve for [tex]\( y_2 \)[/tex], multiply both sides by 2:
[tex]\[ 2 = 5 + y_2 \][/tex]
Then, subtract 5 from both sides:
[tex]\[ y_2 = -3 \][/tex]
Thus, the coordinates of the other end of the line segment are [tex]\( (6, -3) \)[/tex].
