Question

Which sequence of transactions map rectangle A to rectangle B?

A) A dilation centered at the origin followed by a translation.

B) A reflection over the x-axis followed by a dilation and a translation

C) A rotation of 180° about the origin followed by a dilation and a
translation

D) A rotation of 90° about the origin followed by a translation.

Answer 1

The option B) i.e. a reflection over the x-axis followed by a dilation and a translation is the correction option.

Explanation:

Rectangle A with the four vertices.

• Let 'J' be the first vertex (bottom left) with (2, 2) coordinate location.

• Let 'K' be the second vertex (bottom right) with (8, 2) coordinate location.

• Let 'L' be the third vertex (upper left ) with (2, 6) coordinate location.

• Let 'M' be the fourth vertex (upper right ) with (8, 6) coordinate location.

In other words:

• J(2, 2)

• K(8, 2)

• L(2, 6)

• M(8, 6)

After reflection across x - axis, the x coordinate remains the same, but the y-coordinate changes its sign.

i.e. (x, y) → (x, -y)

so

• J(2, 2) → J'(2, -2)

• K(8, 2) → K'(8, -2)

• L(2, 6) → L'(2, -6)

• M(8, 6) → M'(8, -6)

Now, dilate by a scale factor 1/2

The rule of dilation by a scale factor of 1/2 is:

(x, y) → (1/2 x, 1/2 y)

so

J(2, 2) → J'(2, -2) → J''(1, -1)

K(8, 2) → K'(8, -2) → K''(4, -1)

L(2, 6) → L'(2, -6) → L''(1, -3)

M(8, 6) → M'(8, -6) → M''(4, -3)

Then translation: (x, y) → (x - 6, y -1 )

J(2, 2) → J'(2, -2) → J''(1, -1) → J'''(-5, -2)

K(8, 2) → K'(8, -2) → K''(4, -1) → K'''( -2, -2)

L(2, 6) → L'(2, -6) → L''(1, -3) → L'''(-5, -4)

M(8, 6) → M'(8, -6) → M''(4, -3) → M''' (-2, -4)

Therefore, the option B) i.e. a reflection over the x-axis followed by a dilation and a translation is the correction option.