Question

∆ABC rotates 90° clockwise about point P to form ∆A′B′C′.

Answer 1

Assume P(xp,yp), A(xa,ya), etc.
We know that rotation rule of 90° clockwise about the origin is
R_-90(x,y) -> (y,-x)
For example, rotating A about the origin 90° clockwise is
(xa,ya) -> (ya, -xa)
or for a point at H(5,2), after rotation, H'(2,-5), etc.

To rotate about P, we need to translate the point to the origin, rotate, then translate back.  The rule for translation is
T_(dx,dy) (x,y) -> (x+dx, y+dy)

So with the translation set at the coordinates of P, and combining the rotation with the translations, the complete rule is:
T_(xp,yp)  R_(-90)  T_(-xp,-yp) (x,y) 
-> T_(xp,yp)  R_(-90)  (x-xp, y-yp)
-> T_(xp,yp)  (y-yp, -(x-xp))
-> (y-yp+xp, -x+xp+yp)

Example: rotate point A(7,3) about point P(4,2)
=> x=7, y=3, xp=4, yp=2
=> A'(3-2+4, -7+4+2) => A'(5,-1)