Question

Use the image to determine the direction and angle of rotation. Graph of triangle ABC in quadrant 1 with point A at 1 comma 3. A second polygon A prime B prime C prime in quadrant 4 with point A prime at 3 comma negative 1. 90° clockwise rotation 180° clockwise rotation 180° counterclockwise rotation 90° counterclockwise rotation

Answer 1

The direction and angle of rotation is: 90° clockwise rotation

What is the transformation rule?

Some common transformation rules are given as follows:

- 90° clockwise rotation: (x,y) -> (y,-x)

- 90° counterclockwise rotation: (x,y) -> (-y,x)

- 180° clockwise and counterclockwise rotation: (x, y) -> (-x,-y)

- 270° clockwise rotation: (x,y) -> (-y,x)

- 270° counterclockwise rotation: (x,y) -> (y,-x)

A vertex and it's equivalent is given as follows:

A(1,3) and A'(3, -1).

Hence the rule is:

(x,y) -> (y, -x).

Thus, the rule is a 90° clockwise rotation

Answer 2

To determine the direction and angle of rotation needed to move the original triangle ABC to the new polygon A' B' C', we can use a vector-based approach.

First, we can find the translation vector that moves point A to point A':

translation vector = A' - A = (3, -1) - (1, 3) = (2, -4)

Next, we can find the vector that connects point B to point A in the original triangle ABC:

vector AB = B - A

Then, we can rotate this vector using the rotation matrix for the desired angle of rotation:

90° clockwise rotation:
|0 -1| |x| |y'|
|1 0| * |y| = | -x'|

180° clockwise rotation:
|-1 0| |x| |x'|
| 0 -1| * |y| = |y'|

180° counterclockwise rotation:
|-1 0| |x| |-x'|
| 0 -1| * |y| = |-y'|

90° counterclockwise rotation:
| 0 1| |x| |-y'|
|-1 0| * |y| = | x'|

Finally, we can add the translation vector to the rotated vector to get the coordinates of the corresponding point in the new polygon A' B' C':

90° clockwise rotation: B' = A' + (AB rotated 90° clockwise + translation)
180° clockwise rotation: B' = A' + (AB rotated 180° clockwise + translation)
180° counterclockwise rotation: B' = A' + (AB rotated 180° counterclockwise + translation)
90° counterclockwise rotation: B' = A' + (AB rotated 90° counterclockwise + translation)

Using this method, we can find the coordinates of points B' and C' in the new polygon A' B' C'. Then, we can use the coordinates of these points to determine the direction and angle of rotation needed to move the original triangle to the new polygon.