a) Given the triangle ABC, you have to do a counterclockwise 90º rotation of the figure.
To make said rotation you have to invert the coordinates of each point of the figure and invert the sign of the x-coordinate of the image point.
In general:
Preimage point; Image point
P(x,y) → P'(-y,x)
The y-coordinate turns into the x-coordinate and the x-coordinate turns into the y-coordinate.
The x-coordinate of the image point must have the opposite sing as the original one.
So for triangle ABC:
A to A'
(3,-2) → (-(-2),3)= (2,3)
B to B'
(3,-6) → (-(-6),3)= (6,3)
C to C'
(9,-2) → (-(-2),9)= (2,9)
The coordinates for the 90º counterclockwise rotation are A'(2,3), B'(6,3) and C(2,9)
b) Triange A'B'C' was translated a certain number of units, its new position is given as triangle A''B''C'':
A''(-3,-4)
B''(1,-4)
C''(-3,2)
To determine what kind of translation was done, first step is to draw triangle A''B''C'' and compare it to triangle A'B'C':
As you can see in the graphic, triangle A'B'C' was translated horizontally to the left a k number of units and vertically downwards a m number of units.
Horizontal translation
These translations are made over the x-axis, the translation factor k is added (movement to the rigth) or subtracted (movement to the left) from the x-coordinate of each point:
In this case the translation was made to the left, so:
Preimage point; Image point
P(x,y) → P'(x-k,y)
Vertical translation
These translations are made over the y-axis, this means that the translation factor m will be added (↑up) or subtracted (↓down) from the y-coordinates of each point.
For the example, the movement was downwards so we can express it as:
Preimage point; Image point
P(x,y) → P'(x,y-m)
You can unite both movements in the same expression as:
Preimage point; Image point
P(x,y) → P'(x-k,y-m)
Going a little further you can determine the amount of units the figure was translated by comparing a set of points from the preimage and image:
Given A'(2,3) and A''(-3,-4)
For the horizontal movement compare the x-coordinates. We know that to determine the x-coordinate of A'', k units were subtacted from the x-coordinate of A', so:
2-k=-3
-k=-3-2
-k=-5
k=5
For the vertical movement, compare the y-coordinates of both point. We know that m units were subtracted from the y-coordinate of A' to determine the y-coordinate of A'', so:
3-m=-4
-m=-4-3
-m=-7
m=7
This means that the translation rule for A'B'C' → A''B''C'' is (x-5,y-7)