Question

quadrilaterals hijk and abcd are congruent. the side length of each square on the grid is 1 unit.

which of the following sequences of transformations maps hijk onto abcd?

sequence a
a 270° rotation about point j, then a translation 9 units to the right and 8 units down.

sequence b
a reflection over the line hi, then a translation 6 units to the right and 3 units down.

choose 1 answer:
a only sequence a
b only sequence b
c both
d neither

Explanation:

Step1: Analyze Sequence A

A \(270^\circ\) rotation about point \(J\) will rotate quadrilateral \(HIJK\). Then, translating 9 units right and 8 units down. Let's visualize the grid: after rotation, the shape's orientation changes, and then the translation moves it to the position of \(ABCD\). The key is that congruent figures can be mapped via rotations and translations. A \(270^\circ\) rotation (equivalent to \(-90^\circ\) rotation) about \(J\) will align the sides, and then the translation moves it to \(ABCD\)'s location.

Step2: Analyze Sequence B

A reflection over line \(HI\) will flip the quadrilateral \(HIJK\) over \(HI\), changing its orientation to match \(ABCD\) (since \(ABCD\) has a slanted side that can be obtained by reflecting \(HIJK\) over \(HI\)). Then, translating 6 units right and 3 units down. The reflection over \(HI\) (a vertical line, assuming \(HI\) is vertical) will mirror the shape, and then the translation moves it to \(ABCD\)'s position.

Step3: Conclusion

Both sequences, when applied, will map \(HIJK\) onto \(ABCD\) because rotations, reflections, and translations are rigid transformations that preserve congruence, and both sequences' transformations align the congruent quadrilaterals.

Answer:

C. Both