Triangle ABC with vertices A(2,-2) B(2,-7) C(8,-7) Translate 2 units right and 3 units up then Rotate 180 degrees.
translate: (x,y) → (x+2, y+3)
rotate: (x,y) → (-x,-y)
so combined, (x,y) → (-(x+2) , -(y+3))
To translate a figure, you need to add the desired amount to the coordinates of each point. In this case, you want to translate the triangle 2 units to the right and 3 units up. So, let's apply this translation to each vertex of the triangle.
For vertex A(2,-2), you will add 2 units to the x-coordinate (2+2) and 3 units to the y-coordinate (-2+3). So the new coordinates for A' will be (4, 1).
For vertex B(2,-7), you will also add 2 units to the x-coordinate (2+2) and 3 units to the y-coordinate (-7+3). So the new coordinates for B' will be (4, -4).
Finally, for vertex C(8,-7), you will add 2 units to the x-coordinate (8+2) and 3 units to the y-coordinate (-7+3). So the new coordinates for C' will be (10, -4).
Now, to rotate a figure 180 degrees, you need to reflect each point over the origin (0,0). In other words, you will change the sign of both the x and y coordinates.
Applying this rotation to each vertex, the coordinates will become:
Vertex A': (4, 1) after translation.
After rotation: A''(-4, -1)
Vertex B': (4, -4) after translation.
After rotation: B''(-4, 4)
Vertex C': (10, -4) after translation.
After rotation: C''(-10, 4)
So the new triangle after translating 2 units right and 3 units up, then rotating 180 degrees, is triangle A''B''C'' with vertices A''(-4, -1), B''(-4, 4), C''(-10, 4).