Final answer:
The coordinates of A' after rotating point A(-3,4) 180 degrees around point K(-2,-1) are (-1, -6).
Step-by-step explanation:
To rotate point A(-3,4) 180 degrees around point K(-2,-1), we need to find the coordinates of A'.
First, we calculate the distance between A and K:
Distance = sqrt((x2-x1)^2 + (y2-y1)^2) = sqrt((-2-(-3))^2 + (-1-4)^2) = sqrt(1^2 + (-5)^2) = sqrt(1 + 25) = sqrt(26)
Next, we find the x and y components of the vector from K to A:
x = -3 - (-2) = -1
y = 4 - (-1) = 5
Now, we rotate the vector (-1, 5) 180 degrees by changing the signs of both components:
x' = -(-1) = 1
y' = -5
Finally, the coordinates of A' are the coordinates of K plus the rotated vector:
x' = -2 + 1 = -1
y' = -1 + (-5) = -6