Question

What series of transformations to quadrilateral ABCD map the quadrilateral onto ​ quadrilateral EFGH to prove that ABCD≅EFGH ?

Answer 1

B:  a reflection across the x-axis followed by a translation 4 units right.

Comparing the points to their mapped images compares A to E, B to F, C to G, and D to H.  In each point, the y-coordinate switches signs and the x-coordinate is subtracted by 4.

Reflections across the x-axis will switch the sign of the y-coordinate, and reflections across the y-axis will switch the sign of the x-coordinate.  Since the y-coordinates are the ones whose sign changed, the points must have been reflected across the x-axis.

Translations to the left will cause the x-coordinate to increase, and translations to the right will case the x-coordinate to decrease.  Since the x-coordinate of every point was subtracted, this means that the translation must have been to the right.

Answer 2

Answer: The correct series of transformations is (A) a reflection across x-axis, and then a translation 4 units left.

Step-by-step explanation:  We are given to select the correct series of transformations that map quadrilateral ABCD onto the quadrilateral EFGH so that ABCD≅EFGH.

The co-ordinates of the vertices of quadrilateral ABCD are A(2, 4), B(4, 5), C(5, 4) and D(4, 1).

The first transformation will be reflection across the X-axis - Here, the sign before the y-co-ordinate of each vertex will get changed.

After this transformation, the co-ordinates of the vertices become A'(2, -4), B'(4, -5), C'(5, -4) and D'(4, -1). This quadrilateral A'B'C'D' is shown in the attached figure.

The second transformation will be a translation of 4 units left - Here, the x-co-ordinate of each vertex will get decreased by 4 units.

After this transformation, the co-ordinates of the vertices  changes to E(-2, -4), F(0, -5), G(1, -4) and H(0, -1).

Since these are the co-ordinates of the vertices of quadrilateral EFGH, so we prove that ABCD≅EFGH.

Thus, the correct transformations is given by option (A) a reflection across x-axis, and then a translation 4 units left.