A quadrilateral has been translated two units to the left and five units up. Its image vertices are P(1,-5), Q(4, 3), R(-2, 7), S(-3, 1): Determine the coordinates of the pre-image. Explain your thinking.
To determine the pre-image, we need to undo the translation that was applied to the original quadrilateral.
The original quadrilateral had vertices P', Q', R', and S', and we can express their coordinates as follows:
P' = (xP, yP)
Q' = (xQ, yQ)
R' = (xR, yR)
S' = (xS, yS)
Since the quadrilateral was translated left by two units and up by five units, we can express the new coordinates as follows:
P = P' + (-2, 5) = (xP - 2, yP + 5)
Q = Q' + (-2, 5) = (xQ - 2, yQ + 5)
R = R' + (-2, 5) = (xR - 2, yR + 5)
S = S' + (-2, 5) = (xS - 2, yS + 5)
Using the given coordinates of the image vertices, we can set up four equations for xP, yP, xQ, yQ, xR, yR, xS, and yS:
xP - 2 = 1
yP + 5 = -5
xQ - 2 = 4
yQ + 5 = 3
xR - 2 = -2
yR + 5 = 7
xS - 2 = -3
yS + 5 = 1
Solving these equations, we get:
xP = 3, yP = -10
xQ = 6, yQ = -2
xR = 0, yR = 2
xS = -1, yS = -4
Therefore, the pre-image quadrilateral has vertices at P'(3, -10), Q'(6, -2), R'(0, 2), and S'(-1, -4).
To determine the coordinates of the pre-image, we need to reverse the translation process that was applied to the quadrilateral.
The given image vertices are:
P(1, -5), Q(4, 3), R(-2, 7), S(-3, 1)
We know that the quadrilateral has been translated two units to the left and five units up.
To reverse this translation, we need to move the image vertices two units to the right and five units down.
For each vertex, we will subtract 2 from the x-coordinate and subtract 5 from the y-coordinate:
P: x-coordinate: 1 - 2 = -1, y-coordinate: -5 - 5 = -10
Q: x-coordinate: 4 - 2 = 2, y-coordinate: 3 - 5 = -2
R: x-coordinate: -2 - 2 = -4, y-coordinate: 7 - 5 = 2
S: x-coordinate: -3 - 2 = -5, y-coordinate: 1 - 5 = -4
Therefore, the coordinates of the pre-image are:
P'(-1, -10), Q'(2, -2), R'(-4, 2), S'(-5, -4)
These are the original coordinates of the quadrilateral before the translation.