Question

you do 2
point q with coordinate (4, -7) is rotated 270° clockwise about (5, 1). what are the coordinates of its image?

Explanation:

Step1: Translate points

First, translate the center of rotation $(5,1)$ and point $Q(4, - 7)$ so that the center of rotation is at the origin. Let $(x_0,y_0)=(5,1)$ and $(x_1,y_1)=(4,-7)$. The new coordinates of $Q$ after translation are $(x_1 - x_0,y_1 - y_0)=(4 - 5,-7 - 1)=(-1,-8)$.

Step2: Apply rotation rule

A $270^{\circ}$ clock - wise rotation is equivalent to a $90^{\circ}$ counter - clockwise rotation. The rule for a $90^{\circ}$ counter - clockwise rotation about the origin $(0,0)$ is $(x,y)\to(-y,x)$. For the point $(-1,-8)$, after rotation, it becomes $(8,-1)$.

Step3: Translate back

Now, translate the rotated point back to the original coordinate system. Add the coordinates of the original center of rotation $(5,1)$ to the rotated - and - translated point. The new coordinates are $(8 + 5,-1+1)=(13,0)$.

Answer:

$(13,0)$