PQ has endpoints at P (-2,3) and Q (4,1). The center of dilation is (-1,4) and the scale factor is 2. What are the coordinates of the endpoints of P'Q'?
relative to C=(-1,4),
P = (-1,-1)
Q = (5,-3)
So, scaling those coordinates by 2,
P -> (-2,-2)
Q -> (10,-6)
making
P' = (-3,2)
Q' = (9,-2)
That is,
P' = C+2(P-C)
Q' = C+2(Q-C)
To find the coordinates of the endpoints of P'Q', we need to apply the dilation transformation to the original endpoints of PQ.
The dilation transformation involves three steps:
1. Translate the center of dilation to the origin.
2. Scale the coordinates by the scale factor.
3. Translate the origin back to the center of dilation.
Let's go through these steps:
Step 1: Translate the center of dilation to the origin.
To do this, we need to subtract the coordinates of the center of dilation from both endpoints of PQ:
P' (-2 - (-1), 3 - 4) -> P' (-1, -1)
Q' (4 - (-1), 1 - 4) -> Q' (5, -3)
Step 2: Scale the coordinates by the scale factor.
Multiply the coordinates of P' and Q' by the scale factor of 2:
P' (2 * (-1), 2 * (-1)) -> P' (-2, -2)
Q' (2 * 5, 2 * (-3)) -> Q' (10, -6)
Step 3: Translate the origin back to the center of dilation.
Add the coordinates of the center of dilation to both P' and Q':
P' (-2 + (-1), -2 + 4) -> P' (-3, 2)
Q' (10 + (-1), -6 + 4) -> Q' (9, -2)
Therefore, the coordinates of the endpoints of P'Q' are (-3, 2) and (9, -2).