To find an equation of a line perpendicular to JL and passing through K, we need to determine the slope of JL and then use the negative reciprocal of that slope as the slope of the perpendicular line.
The slope of JL can be found using the slope formula: m = (y2 - y1) / (x2 - x1).
For JL, the coordinates of point J are (2, 4) and the coordinates of point L are (6, 8). Applying the formula, we have:
m = (8 - 4) / (6 - 2)
m = 4 / 4
m = 1
Therefore, the slope of JL is 1.
Next, we need to find the coordinates of the midpoint K. The midpoint formula is given by ( (x1 + x2) / 2, (y1 + y2) / 2 ).
For JL, we have:
x1 = 2, x2 = 6
y1 = 4, y2 = 8
Using the midpoint formula, we can calculate the coordinates of point K:
xK = (2 + 6) / 2 = 8 / 2 = 4
yK = (4 + 8) / 2 = 12 / 2 = 6
Therefore, the coordinates of K are (4, 6).
Now that we have the slope of JL (1) and the coordinates of K (4, 6), we can find the slope of the perpendicular line by taking the negative reciprocal of 1.
The negative reciprocal of 1 is -1/1, which simplifies to -1.
So, the slope of the perpendicular line is -1.
Finally, we can use the point-slope form of a line to find the equation of the line perpendicular to JL and passing through K. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) represents a point on the line and m is the slope.
Plugging in the value of K (4, 6) and the slope (-1), we have:
y - 6 = -1(x - 4)
Simplifying, we get:
y - 6 = -x + 4
Rearranging, we obtain the equation of the line:
y = -x + 10
So, the equation of the line perpendicular to JL and passing through K is y = -x + 10.