Question

Denise proposes the following transformation rule for this rotation:

(x,y) ➝ (y+4, -x-2)

Does this rule work? Justify your answer.​

Answer 1

Denise's proposed transformation (x,y) ➝ (y+4, -x-2) is not a valid rotation transformation because it does not follow the standard formulas for rotation around the origin, which also preserve distances to the origin and between points.

Step-by-step explanation:

No, the transformation rule provided by Denise for the rotation, (x,y) ➝ (y+4, -x-2), does not correspond to the standard rotation transformation in a coordinate system. A general rotation by an angle p around the origin transforms a point using the following equations:

x' = x cos p - y sin p

y' = x sin p + y cos p

The proposed transformation instead translates the point by adding constants to the coordinates, which is not a pure rotation. Additionally, swapping x and y and changing signs would represent a reflection and rotation of 90 degrees, not a general rotation by angle p.

To justify the validity of the rotation transformation:

1. Compare it to the standard rotation formulas.
2. Show that the distance of a point to the origin, which is given by the equation r^2 = x^2 + y^2, remains constant in a pure rotation and changes in the given transformation.
3. Discuss whether the rules preserve angles and distances between points, which the standard rotation must do.

Answer 2

The proposed rule correctly works.

Step-by-step explanation:

As the original triangle has the vertices with the coordinates

• A(-6, 6)

• B(-6, 1)

• C(-2, 1)

From the given diagram, the image of the triangle has the vertices

A'(10, 4)

B'(5, 4)

C'(5, 0)

According to the proposed rule, the image should have the following vertices.

(x,y) ➝ (y+4, -x-2)

A(-6, 6) ➝ (6+4, -(-6)-2) = A'(10, 4)

B(-6, 1) ➝ (1+4, -(-6)-2) = B'(5, 4)

C(-2, 1) ➝ (1+4, -(-2)-2) = C'(5, 0)

So, the coordinates of the image of the triangle are correctly drawn as per the proposed rule.

Therefore, the proposed rule correctly works.