Final answer:
The coordinates of point P are (-4.2, 0) and the coordinates of point Q are (-4.2, 9.6). The length of PQ is approximately 9.6 units.
Step-by-step explanation:
To find the coordinates of point P, we need to determine the equation of line AB and then find the x-coordinate where it intersects the x-axis. The equation of line AB can be found using the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept. The slope, m, can be found using the formula: m = (y2 - y1) / (x2 - x1).
Using the given coordinates, we can calculate the slope of line AB:
m = (-6 - 9) / (-5 - (-3)) = -15 / -2 = 7.5
Since point P lies on the x-axis, its y-coordinate is 0. Using the point-slope form: (y - y1) = m(x - x1), we can substitute the values into the equation and solve for x:
0 - 9 = 7.5(x - (-3))
-9 = 7.5(x + 3)
-9 = 7.5x + 22.5
7.5x = -9 - 22.5
7.5x = -31.5
x = -31.5 / 7.5
x = -4.2
Therefore, the coordinates of point P are (-4.2, 0).
To find the coordinates of point Q, we can use the fact that AP:PB = AP:PQ. Since AP:PB = 1:1, this means that the ratio AP:PQ is also 1:1. Therefore, the x-coordinate of point Q is the same as the x-coordinate of point P, which is -4.2. For the y-coordinate of Q, we can use the formula: y = y1 + (y2 - y1) * (x - x1) / (x2 - x1), where (x, y) is point Q and (x1, x2) and (y1, y2) are points A and C, respectively. Substituting the values:
y = 9 + (4 - 9) * (-4.2 - (-3)) / (7 - (-3))
y = 9 + (-5) * (-4.2 + 3) / (7 + 3)
y = 9 + (-5) * (-1.2) / 10
y = 9 + 6 / 10
y = 9 + 0.6
y = 9.6
Therefore, the coordinates of point Q are (-4.2, 9.6).
Finally, to find the length of PQ, we can use the distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2).
PQ = sqrt((-4.2 - (-4.2))^2 + (9.6 - 0)^2)
PQ = sqrt(0^2 + 9.6^2)
PQ = sqrt(0 + 92.16)
PQ = sqrt(92.16)
PQ ≈ 9.6 units