In the diagram shown, /\ABC is drawn along with point D. A(-6,6), B(-6,1), C(-2,1), D(1,-3)

Question

(a) Plot /\A'B'C', the image of /\ABC after a clockwise rotation of 90 degrees about point. Give the coordinates of the new vertices below.

(b) Denise proposes the following transformation rule for this rotation: (x,y) (y+4,-x-2) Does this rule work? Justify your answer.

(c) Verify that /\A'B'C' has the same size and shape as /\ABC. Why is this true?

Answer 1

Assuming the center of rotation is intended to be D(1, -3), the idea is to "translate" the triangle to a center of rotation at the origin, use the formula for a 90 degree clockwise rotation, then translate that image back to having a center of rotation at D.

The formula for the 90 degree clockwise rotation centered at (0,0): (x,y)-> (y, -x).

The first translation:

A' = A - D or (-6, 6) - (1, -3) -> (-7, 9).

Similarly for B and C.

Now rotate A' 90 degrees clockwise:

A'' = (9, 7) (using the rule)

Finally translate that image to a position having a center of rotation at D:

A''' = A'' + D = (9, 7) + (1, -3) = (10, 4). That is the image of A after a 90 degree rotation centered at (1, -3).

Another more generalized approach to a rotation is realizing that it can be accomplished as two reflections for any angle of rotation.

The first reflecting line passes through D and one of the vertices of the triangle.

The second reflecting line is the angle bisector of angle ADA'''. You do not need to find A''' first, but realize that the line passing through D and A''' is determined by rotating the first reflecting line by the desired number of degrees (clockwise or counter). It gets pretty complicated.

This graph shows the given triangle rotated using both the 1st method, and as the result of two reflections.

desmos.com/calculator/ytokyzypln

This graph has some issues with angles of rotation greater than 180 (found in second reflecting line folder).

But this graph does not have that issue:

desmos.com/calculator/3122c40925

The 2nd graph allows you to specify the vertices of the triangle, the center of rotation, and the angle of rotation. Try rotating the preimage -360 or 360 degrees to see that the image maps onto the preimage.