Arhombus has coordinates a(-6, 3), b(-4, 4), c(-2, 3), and d(-4, 2). what are the coordinates of rhombus a′b′c′d′ after a 90° counterclockwise rotation about the origin followed by a translation 3 units to the left and 2 units down?

We are given that the coordinates of the vertices of the rhombus are:
A(-6, 3)
B(-4, 4)
C(-2, 3)
D(-4, 2)

To solve this problem, we must plot this on a graphing paper or graphing calculator to clearly see the movement of the graph. If we transform this by doing a counterclockwise rotation, then the result would be:
A(-6, -3)
B(-4, -4)
C(-2, -3)
D(-4, -2)

And the final transformation is translation by 3 units left and 2 units down. This can still be clearly solved by actually graphing the plot. The result of this transformation would be:

A′(6, -8)
B′(7, -6)
C′(6, -4)
D′(5, -6)

A'(-6, -8)

B'(-7, -6)

C'(-6, -4)

D'(-5, -6)

Step-by-step explanation:

The 90 ° rotation rule counterclockwise on the origin is:

(x, y) -> (-y, x)

Rotating the rhombus in the given coordinates, we have

A(-6, 3) -> A'(-3, -6)

B(-4, 4) -> B'(-4, -4)

C(-2, 3) -> C'(-3, -2)

D(-4, 2) -> D'(-2, -4)  

The translation rule is:

(x, y) -> (x + h, y + k)

Where

h = horizontal change = - 3 (3 units to the left)

k = vertical change = - 2 (2 units down)

When translating, the following coordinates are obtained:

A'(-3, -6) -> A'(-3-3, -6-2) -> A'(-6, -8)

B'(-4, - 4) -> B'(-4-3, -4-2) -> B'(-7, -6)

C'(-3, -2) -> C'(-3-3, -2-2) -> C'(-6, -4)

D'(-2, -4) -> D'(-2-3, -4-2) -> D'(-5, -6)

The final coordinates are:

A'(-6, -8)

B'(-7, -6)

C'(-6, -4)

D'(-5, -6)

Hope this helps!