Question

Polygons ABDC and A′B′D′C′ are shown on the following coordinate grid:

A coordinate grid is shown from positive 6 to negative 6 on the x-axis and from positive 6 to negative 6 on the y-axis. A polygon ABDC is shown with vertex A on ordered pair 2, negative 2, vertex B on ordered pair 4, negative 2, vertex D on ordered pair 5, negative 3 and vertex C on ordered pair 1, negative 3. A polygon A prime B prime D prime C prime is shown with vertex A prime on ordered pair 1, 2 , vertex B prime on ordered pair 1, 4, vertex D prime on ordered pair 2, 5 and vertex C prime on ordered pair 2, 1.

What set of transformations is performed on ABDC to form A′B′D′C′?

A: A 90-degree counterclockwise rotation about the origin followed by a translation 1 unit to the left
B: A translation 1 unit to the left followed by a 90-degree counterclockwise rotation about the origin
C: A 270-degree counterclockwise rotation about the origin followed by a translation 1 unit to the left
D: A translation 1 unit to the left followed by a 270-degree counterclockwise rotation about the origin

(Sorry if its hard to understand)

Answer 1

Answer: A

Explanation:

Let's first review how turning about the origin changes a point's coordinates:

After rotating and translating some points, one can see the following pattern:

• 180° Rotation: (x,y) -> (-x,-y)
• 90° Clockwise Rotation: (x,y) -> (y, -x)
• 90° Counter-clockwise Rotation: (x,y) -> (-y, x)
• Translation a units to the left: (x, y) -> (x-a, y)

Using these patterns will make it much easier to find the correct series of transformations than graphing each one.

A: A 90-degree counterclockwise rotation about the origin followed by a translation 1 unit to the left

Using the patterns found above, this set of transformations would make (x,y) move first to (-y, x), then (-y-1, x).

• A(2,-2) -> A'(1,2)
• B(4,-2) -> B'(1,4)
• D(5,-3) -> D'(2,5)
• C(1,-3) -> C'(2,1)

B: A translation 1 unit to the left followed by a 90-degree counterclockwise rotation about the origin

This would make (x,y) move first to (x-1,y), then to (-y, x-1)

• A(2,-2) -> A'(2, 1)
• B(4,-2) -> B'(2,3)
• D(5,-3) -> D'(3,4)
• C(1,-3) -> C'(3,0)

C: A 270-degree counterclockwise rotation about the origin followed by a translation 1 unit to the left

This would make (x,y) move first to (y, -x), then (y-1, -x)

• A(2,-2) -> A'(-3, -2)
• B(4,-2) -> B'(-3,-4)
• D(5,-3) -> D'(-4, -5)
• C(1,-3) -> C'(-4, -1)

D: A translation 1 unit to the left followed by a 270-degree counterclockwise rotation about the origin

This would make (x,y) move first to (x-1,y), then (y, -x+1)

• A(2,-2) -> A'(-2, -1)
• B(4,-2) -> B'(-2, -3)
• D(5,-3) -> D'(-3, -4)
• C(1,-3) -> C'(-3, 0)

Therefore, the answer is A, as only it correctly matches to polygon A'B'D'C'.