A graphic designer uses a coordinate plane to design a logo containing a triangular shape.… — ACT

A graphic designer uses a coordinate plane to design a logo containing a triangular shape. The final image of the triangle has vertices at $(3, -1)$ , $(5, -3)$ , and $(2, -5)$ . The designer created this final image by performing a $90^\circ$ counterclockwise rotation of the original triangle about the origin, followed by a translation of $4$ units to the right and $3$ units down. What are the coordinates of the vertices of the original triangle?

$(2, 1)$ , $(0, -1)$ , and $(-2, 2)$ Answer
B
$(-2, -1)$ , $(0, 1)$ , and $(2, -2)$
C
$(-4, -7)$ , $(-6, -9)$ , and $(-8, -6)$
D
$(5, 0)$ , $(7, 2)$ , and $(9, -1)$
E
$(-1, -2)$ , $(1, 0)$ , and $(-2, 2)$

Answer

The coordinates of the vertices of the original triangle are

(2, 1)

(0, -1)

, and

(-2, 2)

The correct answer is obtained by working backward from the final image. First, undo the translation of

4

units right and

3

units down by translating the final vertices

4

units left and

3

units up. This yields intermediate vertices

(-1, 2)

(1, 0)

, and

(-2, -2)

. Second, undo the

90^\circ

counterclockwise rotation by rotating these intermediate vertices

90^\circ

clockwise about the origin. The algebraic rule for a

90^\circ

clockwise rotation is

(x, y) \rightarrow (y, -x)

. Applying this rule to

(-1, 2)

(1, 0)

, and

(-2, -2)

gives the original coordinates

(2, 1)

(0, -1)

, and

(-2, 2)

Step-by-Step Solution

Identify the two transformations in reverse order and define their inverse operations.

The final transformation was a translation of

4

units to the right and

3

units down, so the first step in working backward is a translation of

4

units to the left and

3

units up. The initial transformation was a

90^\circ

counterclockwise rotation about the origin, so the second step in working backward is a

90^\circ

clockwise rotation about the origin.

To find the pre-image, we must apply the inverse of each transformation in the reverse order of how they were originally applied.

Apply the inverse translation of

4

units left and

3

units up to the final image vertices:

(3, -1)

(5, -3)

, and

(2, -5)

The intermediate vertices are:

A'(3, -1) \rightarrow A_1(3 - 4, -1 + 3) = (-1, 2)

;

B'(5, -3) \rightarrow B_1(5 - 4, -3 + 3) = (1, 0)

;

C'(2, -5) \rightarrow C_1(2 - 4, -5 + 3) = (-2, -2)

Undoing a translation of

(+4, -3)

requires subtracting

4

from the

x

-coordinates and adding

3

to the

y

-coordinates.

Apply the inverse rotation, a

90^\circ

clockwise rotation about the origin, to the intermediate vertices.

The rule for a

90^\circ

clockwise rotation about the origin is

(x_1, y_1) \rightarrow (y_1, -x_1)

. Applying this to the intermediate vertices yields:

A_1(-1, 2) \rightarrow A(2, 1)

;

B_1(1, 0) \rightarrow B(0, -1)

;

C_1(-2, -2) \rightarrow C(-2, 2)

90^\circ

clockwise rotation is the inverse of a

90^\circ

counterclockwise rotation, which reverses the coordinates and changes the sign of the new

y

-coordinate.

Key Concept

Finding the pre-image of a figure under composite transformations in the coordinate plane by applying inverse transformations in reverse order.

Question

Answer

Step-by-Step Solution

Key Concept

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