Final answer:
The coordinates of P are (6, 5), and the coordinates of Q are (1, 0). A, Q, and B are proven to be collinear by showing they have equal slopes. However, PQLAB does not hold due to angle and orientation differences. The area of triangle AAPB is 24√2 square units.
Step-by-step explanation:
Coordinates of P and Q
The student has provided coordinates for points A and B and transformations that were applied to each point. To find P, the point after rotating A clockwise about the origin by 270°, we perform the rotation which results in flipping the coordinates and changing the sign of the now x-coordinate (previously y-coordinate). Therefore, the coordinates of P are (6, 5).
To find Q, the point after translating B down by 2 units and to the left by 2 units, we subtract 2 from both the x and y coordinates of B. Hence, the coordinates of Q are (1, 0).
Proving A, Q, and B are Collinear
To prove that A, Q, and B are collinear, we check the slopes of line segments AB and AQ to see if they are equal. The slope of AB is (2 - (-6)) / (3 - (-5)) = 8/8 = 1. The slope of AQ is (0 - (-6)) / (1 - (-5)) = 6/6 = 1. Since both slopes are equal, A, Q, and B are collinear.
Proving PQLAB
Point Q is a translation of B, and since translations preserve angle and distance, angle QLB is equal to angle AQB. Point P is derived from a rotation of A, preserving distances but changing orientation; therefore, PL is not parallel to AB, and angles at L are not equal. Therefore, PQLAB does not hold.
Finding the Area of AAPB
To find the area of triangle AAPB, we use the coordinates to determine the base AB and the height AP. The distance AB is √((3 - (-5))^2 + (2 - (-6))^2) = √(64 + 64) = √128 = 8√2 units. The height AP can be found by taking the absolute value of the x-coordinate of P (as A and P both lie on a vertical line when x=0 is the base), which is 6 units. Using the formula for the area of a triangle (1/2 * base * height), we get 1/2 * 8√2 * 6 = 24√2 square units.