To find the value of x, we first need to find the equation of the line L that goes through the points (3, 7) and (5, 12).
The slope of this line is:
m = (y2 - y1) / (x2 - x1) = (12 - 7) / (5 - 3) = 5/2
We can use the point-slope form of the equation of a line to find the equation of L:
y - y1 = m(x - x1)
Using the point (3, 7):
y - 7 = (5/2)(x - 3)
Simplifying:
y = (5/2)x - (1/2)
Since L is perpendicular to this line, its slope is the negative reciprocal of 5/2:
m' = -2/5
Using the point-slope form of the equation of a line again, we can find the equation of L:
y - 13 = (-2/5)(x - 0)
Simplifying:
y = (-2/5)x + 13
To find the value of x when this line passes through (x, 21), we can substitute y = 21 into the equation:
21 = (-2/5)x + 13
Solving for x:
x = (21 - 13) / (-2/5) = -4.375
Therefore, the value of x is approximately -4.375.