A line segment has endpoints C(-3,-6) and D(-6,-4). Point P is such that PC is perpendicular to CD. Determine the coordinates of P if P is on the x-axis.
P is (0,T)
slope of CD = 2/-3 = -2/3
slope of PC is then 3/2
so PC is
y = (3/2)x + b
at C, x = -3 and y = -6
-6 = 1.5 (-3) + b
b = -6 + 4.5 = -1.5
so
y = 1.5 x - 1.5
on the x axis when y = 0
x = 1
sp
P at (1,0)
To find the coordinates of point P on the x-axis, we need to find the point where the line PC is perpendicular to the line CD.
First, let's find the equation of the line segment CD using the two given endpoints, C(-3,-6) and D(-6,-4).
The equation of a line passing through two points (x₁,y₁) and (x₂,y₂) is given by:
y - y₁ = (y₂ - y₁) / (x₂ - x₁) * (x - x₁)
Using the coordinates of C(-3,-6) and D(-6,-4):
y - (-6) = (-4 - (-6)) / (-6 - (-3)) * (x - (-3))
Simplifying the equation:
y + 6 = 2/3 * (x + 3)
Next, we need to find the slope of the line CD. The slope of a line passing through two points (x₁,y₁) and (x₂,y₂) is given by:
slope = (y₂ - y₁) / (x₂ - x₁)
Using the coordinates of C(-3,-6) and D(-6,-4):
slope_CD = (-4 - (-6)) / (-6 - (-3))
= 2 / (-3)
= -2/3
Since the perpendicular line has the negative reciprocal slope, the slope of the line PC would be the negative inverse of the slope_CD:
slope_PC = -1 / slope_CD
= -1 / (-2/3)
= 3/2
Now, since point P lies on the x-axis, its y-coordinate will be 0. Using the slope-intercept form of the line y = mx + b, we substitute the slope and the coordinates (x,y) of point P (where y = 0) into the equation:
0 = (3/2) * x + b
Solving for b:
0 - (3/2) * x = b
b = - (3/2) * x
So, the equation of line PC is:
y = (3/2) * x - (3/2) * x
y = 0
Therefore, point P lies on the x-axis. Its y-coordinate is 0 and the x-coordinate can be determined using the equation of line PC. We know that the line passes through point C (-3,-6), so we can substitute these coordinates into the equation:
0 = (3/2) * x - (3/2) * (-3)
0 = (3/2) * x + 9/2 - 9/2
0 = (3/2) * x
To solve for x, we have:
0 = (3/2) * x
x = 0
Therefore, the coordinates of point P are (0,0) when P lies on the x-axis.