△ABC is mapped to △A′B′C′ using each of the given rules.

Response:

Congruent:  (x, y)→(x+3, y-4); (x, y)→(-x, -y)

Not Congruent:  (x, y)→(3x, 3y); (x, y)→(0.4x, 0.4y); (x, y)→(x/3, y/3)

Explanation in steps:

Transformations that yield congruent figures include translations, reflections, and rotations. In contrast, dilations result in figures that are not congruent.

The first transformation, (x, y)→(x+3, y-4), represents a translation 3 units right and 4 units down, leading to congruent figures since it merely shifts the figure.

The second transformation, (x, y)→(3x, 3y), indicates a dilation by a factor of 3, which enlarges the figure and therefore makes it non-congruent.

The third transformation, (x, y)→(0.4x, 0.4y), involves a dilation by a factor of 0.4, which shrinks the figure, leading to non-congruence.

The fourth transformation, (x, y)→(x/3, y/3), results in a dilation by a factor of 1/3, also shrinking the figure and rendering it non-congruent.

The fifth transformation, (x, y)→(-x, -y), reflects the figure, maintaining its size while modifying its orientation, thus ensuring congruence.

Response:- Congruent =  (x, y)→(x+3, y-4)

                                        (x, y)→(-x, -y)

               Not Congruent=  (x, y)→(3x, 3y)

                                            (x, y)→(0.4x, 0.4y)

                                             (x, y)→(x/3, y/3)

Explanation:

1.The transformation (x, y)→(x+3, y-4) is a translation that moves the x coordinates of points of ΔABC 3 units right and the y coordinate 4 units downward. Being a rigid transformation, it doesn't alter the figure's size but shifts every point to create its image, making △ABC congruent to △A′B′C′.

2.The transformation (x, y)→(3x, 3y) signifies a dilation by a factor of 3. Following this dilation, the image changes size, so △ABC becomes non-congruent to △A′B′C′.

3.The transformation (x, y)→(0.4x, 0.4y) indicates a dilation by a factor of 0.4. This also changes the image size, resulting in △ABC being non-congruent to △A′B′C′.  

4.The transformation (x, y)→(x/3, y/3) signifies a dilation by a factor of 1/3, leading to a change in size and making △ABC non-congruent to △A′B′C′.

5.The transformation (x, y)→(-x, -y) is a reflection that retains the figure's size while changing its orientation, hence making △ABC congruent to △A′B′C′.