On a coordinate plane, 2 trapezoids are shown. The first trapezoid has points A (negative 1, negative 2), B (1, negative 2), C (2, negative 5), and D (negative 2, negative 5). The second trapezoid has points A prime (negative 4, negative 1), B prime (negative 2, negative 1), C prime (negative 1, negative 4), and D prime (negative 5, negative 4). Which statements are true about trapezoid ABCD and its translated image, A'B'C'D'? Select two options. The rule for the translation can be written as T–3, 1(x, y). The rule for the translation can be written as T–1, 3(x, y). The rule for the translation can be written as (x, y) → (x + 1, y – 3). The rule for the translation can be written as (x, y) → (x – 3, y + 1). Trapezoid ABCD has been translated 3 units to the right and 1 unit up.

The rule for the translations can be written as [tex]\mathbf{T_{-3,1}(x,y) }[/tex]
The rule for translations can be written as [tex]\mathbf{(x,y) \to (x - 3,y+1)}[/tex]

From the question, we have the following vertices of the first and the second triangles.

[tex]\mathbf{A = (-1,-2)}[/tex]

[tex]\mathbf{B = (1,-2)}[/tex]

[tex]\mathbf{C = (2,-5)}[/tex]

[tex]\mathbf{D = (-2,-5)}[/tex]

[tex]\mathbf{A' = (-4,-1)}[/tex]

[tex]\mathbf{B' = (-2,-1)}[/tex]

[tex]\mathbf{C' = (-1,-4)}[/tex]

[tex]\mathbf{D' = (-5,-4)}[/tex]

Using points A and A', the translation rule

[tex]\mathbf{(x,y) = A - A'}[/tex]

This gives

[tex]\mathbf{(x,y) = (-1,-2) - (-4,-1)}[/tex]

[tex]\mathbf{(x,y) = (-1+4,-2+1)}[/tex]

[tex]\mathbf{(x,y) = (3,-1)}[/tex]

This means that, the translation rule is:

[tex]\mathbf{(x,y) \to (x - 3,y+1)}[/tex]

or

[tex]\mathbf{T_{-3,1}(x,y) }[/tex]