Question

Which of the steps will cause the rectangle to map onto itself?A. Reflect across the x-axis, then shift 9 units upB. Reflect across the x-axis, then shift 10 units upООООC. Reflect across the x-axis, then shift 6 units upD. Reflect across the x-axis, then shift 4 units up

Answer 1

We need to identify the coordinates of each of the vertex of the rectangle:

We have:

A(1, 3), B(1, 6), C(8, 6), D(8, 3)

If we reflect the rectangle across the x-axis, we have that its coordinates are:

(x, y) ---> (x, -y) (these are the coordinates of the image.

Then, we have:

A(1, 3) ---> A'(1, -3)

B(1, 6) ---> B'(1, -6)

C(8, 6) ---> C'(8, -6)

D(8, 3) ---> D'(8, -3)

If we shift the reflection 6 units up, we have that:

(x, y) ---> (x, y + 9)

Then, we have:

A'(1, -3) ---> (1, -3 + 9) ---> A" (1, 6) (B in the original image.)

B'(1, -6) ---> (1, -6 + 9) ---> B" (1, 3) (A in the original image.)

C'(8, -6) ---> (8, -6 + 9) ---> C"( 8, 3) (D in the original image.)

D'(8, -3) ---> (8, -3 + 9) ---> D" (8, 6) (C in the original image.)

Now, A = B'', B = A'', C = D'' and D = C''

Note that after reflection point A becomes A'. However, after shifting upwards, point A' takes the position of B (in the original image). So point A' does not return to (1, 3) but shifts upwards to (1, 6).

Therefore, the correct option is option A: Reflect across the x-axis, then shift 9 units up.

The graph of the reflection (and this can help to understand better the explanation):