Question

Describe in words a sequence of transformations that maps ∆ABC to ∆A'B'C'

Transformation #1

Transformation #


Write an ordered-pair rule for each transformation in the sequence:

Transformation #1

Transformation #2

Answer 1

Answer:

1. rotation 90° CCW; (x, y) ⇒ (-y, x)
2. translation 2 left, 4 up; (x, y) ⇒ (x-2, y+4)

Step-by-step explanation:

There are many ways the figure could be transformed to get from one to the other. The orientation of both figures is ABC is CCW, so if reflection is used, an even number of reflections are needed. The direction of AB is "east" and the direction of A'B' is "north", so we know the transformations have to result in a rotation 90° counterclockwise.

Quick and Easy

Since point A is at the origin, rotation 90° CCW about the origin leaves point A where it is, and aligns the AB segment on the y-axis. From there, the translation up and left is easy to see.

Other Alternatives

There are other transformations that could be used, including rotation about (-3, 1). This latter transformation could do the whole thing in one step. (Your problem requires 2 steps, so this is not a suitable answer.)

Answer 2

Answer:  The required sequence of transformations is

Transformation 1 : a rotation by 90 degrees in the counterclockwise direction, (x, y) ⇒  (-y, x).

Transformation 2 : a translation by 2 units left and 4 units up, (x, y) ⇒ (x-2, y+4).

Step-by-step explanation:  We are given to describe in words a sequence of transformations that maps ∆ABC to ∆A'B'C'.

From the figure, we note that

the co-ordinates of the vertices of triangle ABC are A(0, 0), B(3, 0) and C(2, 1).

And, the co-ordinates of the vertices of triangle A'B'C' are A'(-2, 4), B'(-2, 7) and C'(-3, 6).

We see that if triangle ABC is rotated 90 degrees in counterclockwise direction, then its co-ordinates changes according to the following rule :

(x, y)  ⇒  (-y, x).

That is

A(0, 0)  ⇒  (0, 0),

B(3, 0)   ⇒ (0, 3),

C(2, 1)    ⇒ (-1, 2).

Now, if the vertices of the rotated triangle are translated  2 units left and 4 units up, then

(x, y)  ⇒  (x-2, y+4).

That is, the final co-ordinates after rotation and translation will be

(0, 0)  ⇒  (0-2, 0+4) = (-2, 4),

(0, 3)   ⇒ (0-2, 3+4) = (-2, 7),

(-1, 2)   ⇒ (-1-2, 2+4) = (-3, 6).

We see that the final co-ordinates are the co-ordinates of the vertices of triangle A'B'C'.

Thus, the required sequence of transformations is

Transformation 1 : a rotation by 90 degrees in the counterclockwise direction, (x, y) ⇒  (-y, x).

Transformation 2 : a translation by 2 units left and 4 units up, (x, y) ⇒ (x-2, y+4).