Given ΔABC with A(1, 1), B(7, -2), and C(1, -2) and if the  is rotated 90° about the (-2, -4), the new location of the B'

To find the new location of point B' after a 90° rotation about the point (-2, -4), we can use the rotation formula:

New x = xcosθ - ysinθ New y = xsinθ + ycosθ

Here, θ is the angle of rotation, which is 90° or π/2 in radians. The coordinates of point B are (7, -2) and the center of rotation is (-2, -4).

However, before applying the rotation, we need to shift our point B relative to the center of rotation. So, we subtract the coordinates of the center of rotation from the coordinates of point B:

Shifted B = B - center of rotation = (7 - (-2), -2 - (-4)) = (9, 2)

Now we can apply the rotation:

B'x = 9cos(π/2) - 2sin(π/2) = 0 - 2 = -2 B'y = 9sin(π/2) + 2cos(π/2) = 9 + 0 = 9

Finally, we shift our point B' back by adding the coordinates of the center of rotation:

Final B' = B' + center of rotation = (-2 + (-2), 9 + (-4)) = (-4, 5)

So, the new location of point B' after a 90° rotation about the point (-2, -4) is (-4, 5).