Question

date
period

1. name a transformation that takes figure a to figure b. name a transformation that takes figure b to figure c.

1. what is one sequence of transformations that takes figure a to figure c? explain how you know.

are you ready for more?
experiment with some other ways to take figure a to figure c. for example, can you do it with...

• no rotations?
• no reflections?
• no translations?

Explanation:

Question 1

Part 1: Transformation from A to B

Step1: Analyze A to B

Figure A and B have same shape, size, orientation. Distance between corresponding points is constant (horizontal shift). So translation (slide) right.

Step2: Analyze B to C

Figure B and C: B can be translated down, then rotated (or rotated then translated) to match C. Or reflection? Wait, B to C: Let's see orientation. B's sides: if we translate B down a few units and rotate (or reflect? Wait, maybe translation and rotation, or translation and reflection. Wait, more accurately: From B to C, first translate down (say 3 units) and then rotate 90 degrees (or reflect? Wait, maybe translation and rotation. Alternatively, translation and reflection. But let's check coordinates. Alternatively, translation (down) and rotation. But more precise: Transformation from B to C: translation (down) and rotation (or reflection). Wait, maybe translation (down) and then rotation, or a rotation combined with translation. But the key is: A to B is translation (since same orientation, just shifted right). B to C: let's see, B's shape: if we translate B down (say 4 units) and then rotate 90 degrees clockwise, or reflect? Wait, maybe translation (down) and rotation. Alternatively, a rotation and translation. But the standard: A to B is translation (horizontal shift). B to C: translation (down) and rotation (or reflection). But let's confirm:

For A to B: same shape, size, orientation. So translation (slide) to the right (number of units: count the grid. Let's say 4 units right).

For B to C: B can be translated down (say 3 units) and then rotated 90 degrees clockwise (or counterclockwise), or reflected. Alternatively, a rotation about a point and translation. But the main is: A to B: translation. B to C: rotation (and translation) or reflection (and translation). But more accurately, from B to C: first translate down (maybe 3 units) and then rotate 90 degrees, or a rotation combined with translation.

Part 2: Transformation from B to C

Step1: Observe B and C

B and C: different orientation. So need rotation (and translation). Let's say translate B down (e.g., 3 units) and then rotate 90 degrees clockwise (or counterclockwise), or reflect. Alternatively, rotate B 90 degrees and then translate. But the key is: A to B is translation (right). B to C is translation (down) and rotation (or reflection).

Question 2: Sequence from A to C

Step1: A to B

Translate Figure A to the right (same as Q1) to get Figure B.

Step2: B to C

Then, translate Figure B down (say 3 units) and rotate 90 degrees clockwise (or other rotation/reflection) to get Figure C. Alternatively, first rotate A, then translate, but the simplest is A→B (translation) then B→C (translation + rotation). We know because A and B are congruent (same shape, size, orientation) so translation. B and C are congruent (same shape, size) so transformation preserving congruence (translation + rotation/reflection) works.

Are you ready for more?

No rotations?

Yes: Translate A to B (translation), then translate B down and reflect (if reflection is allowed, but no rotations. So translation (A→B), then translation (down) and reflection (instead of rotation) to get from B to C. Since reflection doesn't use rotation, this works.

No reflections?

Yes: Translate A to B (translation), then translate B down and rotate (since no reflections, use rotation). So translation (A→B), translation (down) and rotation (B→C) – no reflections, just translations and rotation.

No translations?

Yes: Rotate A first (to match orientation of C) and then reflect (or rotate) with no translation? Wait, A and C are not in same position. Wait, no translations: so we need to rotate A around a point to get to C's position. Let's see: Find a center of rotation such that rotating A around it maps to C. So rotation about a point (no translation) – yes, possible. Rotate A 90 degrees clockwise around a point that shifts it to C's position. So yes, can do with no translations (just rotation).

Final Answers (Question 1)

1. Transformation from A to B: Translation (e.g., 4 units to the right).

Transformation from B to C: Translation (e.g., 3 units down) and Rotation (e.g., 90° clockwise) (or Translation + Reflection, depending on grid).

Final Answers (Question 2)

One sequence: Translate Figure A to the right (to get B), then translate B down and rotate 90° clockwise (to get C). We know because translations and rotations preserve congruence, and A→B (same shape/orientation) is translation; B→C (same shape, different orientation) needs translation + rotation.

Final Answers (Are you ready for more?)

• No rotations? Yes: Translate A→B, then translate B→C with reflection (no rotation).
• No reflections? Yes: Translate A→B, then translate B→C with rotation (no reflection).
• No translations? Yes: Rotate A around a point to map to C (no translation).

Answer:

Step1: A to B

Translate Figure A to the right (same as Q1) to get Figure B.

Step2: B to C

Then, translate Figure B down (say 3 units) and rotate 90 degrees clockwise (or other rotation/reflection) to get Figure C. Alternatively, first rotate A, then translate, but the simplest is A→B (translation) then B→C (translation + rotation). We know because A and B are congruent (same shape, size, orientation) so translation. B and C are congruent (same shape, size) so transformation preserving congruence (translation + rotation/reflection) works.

Are you ready for more?

No rotations?

Yes: Translate A to B (translation), then translate B down and reflect (if reflection is allowed, but no rotations. So translation (A→B), then translation (down) and reflection (instead of rotation) to get from B to C. Since reflection doesn't use rotation, this works.

No reflections?

Yes: Translate A to B (translation), then translate B down and rotate (since no reflections, use rotation). So translation (A→B), translation (down) and rotation (B→C) – no reflections, just translations and rotation.

No translations?

Yes: Rotate A first (to match orientation of C) and then reflect (or rotate) with no translation? Wait, A and C are not in same position. Wait, no translations: so we need to rotate A around a point to get to C's position. Let's see: Find a center of rotation such that rotating A around it maps to C. So rotation about a point (no translation) – yes, possible. Rotate A 90 degrees clockwise around a point that shifts it to C's position. So yes, can do with no translations (just rotation).

Final Answers (Question 1)

1. Transformation from A to B: Translation (e.g., 4 units to the right).

Transformation from B to C: Translation (e.g., 3 units down) and Rotation (e.g., 90° clockwise) (or Translation + Reflection, depending on grid).

Final Answers (Question 2)

One sequence: Translate Figure A to the right (to get B), then translate B down and rotate 90° clockwise (to get C). We know because translations and rotations preserve congruence, and A→B (same shape/orientation) is translation; B→C (same shape, different orientation) needs translation + rotation.

Final Answers (Are you ready for more?)

• No rotations? Yes: Translate A→B, then translate B→C with reflection (no rotation).
• No reflections? Yes: Translate A→B, then translate B→C with rotation (no reflection).
• No translations? Yes: Rotate A around a point to map to C (no translation).