Question

PPls help with this geometry question. I really could use step by step solution so i can teach my son. Thanks

Answer 1

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Answer:

  (a) rotation 90° clockwise and translation 5 units down

  (b) (x, y) ⇒ (y, -x -5) . . . . does both transformations

Step-by-step explanation:

The rigid transformations we're concerned with are ...

• rotation about a point
• reflection across a point or line
• translation

These transformations are called "rigid transformations" because you can cut the figure out of cardboard and rotate it or move it somewhere else (translate it) or flip it over (reflect it) without changing any dimensions.

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If you consider what happens to "left" and "right" or "clockwise" and "counterclockwise" when you look in a mirror, you realize they are reversed. That is, reflection changes the left/right (or clockwise/counterclockwise) orientation of a figure.

One way to tell if that is the case here is to consider the order of side lengths, shortest to longest. In Figure A, that order is clockwise. It is the same in Figure B. This means there is no reflection transformation involved in getting from A to B.

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Rotation

If you look at the directions of the line segments, you see that the shortest in Figure A is horizontal, and the shortest of Figure B is vertical. Changing the orientation in this way requires a rotation. The shortest angle of rotation that gets from A to B is a rotation of 90° clockwise.

When rotation occurs, the center of rotation must be specified. Usually it is considered to be the origin (where the x- and y-axes cross), unless specified otherwise. If you trace the axes and Figure A on another (thin) piece of paper, stick a pin through the two graphs at the origin and rotate your tracing 90° clockwise, you will see that Figure A ends up in the first quadrant. The point nearest the origin will have rotated to (2, 2), and the point farthest from the origin will have rotated to (5, 4).

You will notice that all of the points on the +x axis are ones that were on the +y axis. Similarly, all of the points now on the +y axis are ones that were on the -x axis.

We can describe this 90° clockwise rotation by the effect it has on the point coordinates:

  (x, y) ⇒ (y, -x) . . . . . rotation 90° clockwise

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Translation

In this case, it happens that the x-coordinates of the rotated Figure A are the same as those of Figure B. However, the rotated figure needs to be moved down 5 units to make it coincide with Figure B. This movement without changing the directions of any line segments is called "translation."

Translation horizontally is accomplished by adding a distance to the x-value. Translation vertically is accomplished by adding the distance to the y-value. This should be obvious when you consider what the x- and y-coordinates represent. Here, we want to move the rotated figure down 5 units, so the translation is described by ...

  (x, y) ⇒ (x, y -5)

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(a) Figure A is transformed to Figure B by rotating it 90° clockwise and translating it 5 units down.

(b) The transformation mapping rules are ...

  (x, y) ⇒ (y, -x) . . . . . rotation 90° clockwise

  (x, y) ⇒ (x, y -5) . . . . translation 5 units down

The single rule that performs both mappings is ...

  (x, y) ⇒ (y, -x -5)

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Additional comment

The attached shows the rotation and translation transformations separately. When an original point is transformed, we often add an apostrophe to its letter designator. The second transformation adds another apostrophe. We read the names as "A" and "A prime" and "A double prime".

It really is a good idea to play with these figures drawn on paper and using a pin in the center of rotation—at least until you are able to visualize the result confidently without that manipulative aid.