Dilations practice
1. Given triangle ABC with coordinates A(1, 2), B(4, 3), and C(3, 5), dilate the triangle by a scale factor of 2 with respect to the origin.
To dilate the triangle by a scale factor of 2 with respect to the origin, we multiply the coordinates of each point by 2:
A' = (1 * 2, 2 * 2) = (2, 4)
B' = (4 * 2, 3 * 2) = (8, 6)
C' = (3 * 2, 5 * 2) = (6, 10)
Therefore, the new coordinates of triangle ABC after dilation by a scale factor of 2 with respect to the origin are A'(2, 4), B'(8, 6), and C'(6, 10).
2. Given rectangle PQRS with coordinates P(1, 1), Q(1, 4), R(4, 4), and S(4, 1), dilate the rectangle by a scale factor of 3 with respect to point (2, 2).
To dilate the rectangle by a scale factor of 3 with respect to point (2, 2), we first translate the rectangle so that the center of dilation is at the origin. The translation vector is (-2, -2):
P'(1 - 2, 1 - 2) = (-1, -1)
Q'(1 - 2, 4 - 2) = (-1, 2)
R'(4 - 2, 4 - 2) = (2, 2)
S'(4 - 2, 1 - 2) = (2, -1)
Now, we dilate the translated rectangle by a scale factor of 3:
P'' = (-1 * 3, -1 * 3) = (-3, -3)
Q'' = (-1 * 3, 2 * 3) = (-3, 6)
R'' = (2 * 3, 2 * 3) = (6, 6)
S'' = (2 * 3, -1 * 3) = (6, -3)
Finally, we translate the dilated rectangle back to its original position by adding the translation vector back to each point:
New P = (-3 + 2, -3 + 2) = (-1, -1)
New Q = (-3 + 2, 6 + 2) = (-1, 8)
New R = (6 + 2, 6 + 2) = (8, 8)
New S = (6 + 2, -3 + 2) = (8, -1)
Therefore, the new coordinates of rectangle PQRS after dilation by a scale factor of 3 with respect to point (2, 2) are P(-1, -1), Q(-1, 8), R(8, 8), and S(8, -1).