Design your own real-world scenario involving a geometric figure and its transformations. Model the pre-image and image on a coordinate plane. In your final answer, include the real-world scenario, written in complete sentences, the transformations that map the pre-image onto its image, and all calculations involved for the equations of the corresponding lines or the coordinates for the corresponding vertices. Also, please include a sketch of the pre-image and the image.
makeup a geometric figure scenario
just the scenario? try... having to move a geometric figure(which could be fabric)... onto a piece of paper for a craft. you could say how many inches it needs to move up/down and left/right.
(but it needs to still have coordinates)
Real-World Scenario: Rearranging Furniture in a Living Room
In this scenario, we will consider a real-world situation of rearranging furniture in a living room. We have an L-shaped sofa and a rectangular coffee table, and we want to create a new seating arrangement. To achieve this, we will use transformations to map the pre-image furniture onto its image.
1. Pre-Image:
Let's place the L-shaped sofa and the rectangular coffee table in their original positions on a coordinate plane. The L-shaped sofa has one section located at (2, 3) and the other section extends horizontally from there. The length of the sofa is 6 units, and its height is 3 units. The rectangular coffee table is placed at the coordinates (4, 8), with a length of 4 units and a width of 2 units. We will label the L-shaped sofa as ABCD, with A at the bottom left, and the coffee table as EFGH.
Sketch of Pre-Image:
```
(2, 3) A__________B
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| D |
(4, 8) E_____|__________|
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|__________|
```
2. Transformations:
To create the new seating arrangement, we will apply the following transformations:
a) Translation:
We will move the L-shaped sofa and the coffee table 2 units to the right and 5 units upwards. This will result in a translation transformation for both shapes.
b) Reflection:
Next, we will reflect the L-shaped sofa and the coffee table about the y-axis. This will give us a mirror image of the shapes.
c) Rotation:
Finally, we will rotate the L-shaped sofa and the coffee table 90 degrees counterclockwise around the origin. This transformation will change their orientation.
3. Image:
Applying these transformations to the pre-image furniture, we obtain the image. Let's calculate the coordinates for the image:
a) Translation:
The new coordinates for the L-shaped sofa after the translation will be (4, 8) for point A, (6, 8) for point B, (4, 10) for point C, and (6, 10) for point D. Similarly, the new coordinates for the coffee table after the translation will be (6, 13) for point E, (8, 13) for point F, (6, 15) for point G, and (8, 15) for point H.
b) Reflection:
The reflection about the y-axis does not change the coordinates of the shapes. So the coordinates for the reflection remain the same.
c) Rotation:
The rotation by 90 degrees counterclockwise around the origin swaps the x and y coordinates for each point. Therefore, the new coordinates for the L-shaped sofa after the rotation will be (8, 4) for point A, (8, 6) for point B, (10, 4) for point C, and (10, 6) for point D. The new coordinates for the coffee table after the rotation will be (13, 6) for point E, (13, 8) for point F, (15, 6) for point G, and (15, 8) for point H.
Sketch of Image:
```
(8, 4) A__________B
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| D |
(13, 6) E_____|__________|
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|__________|
```
In conclusion, we have successfully rearranged the furniture, consisting of an L-shaped sofa and a rectangular coffee table, in a living room. We used a combination of translation, reflection, and rotation transformations to map the pre-image furniture onto its image. The calculations involved finding the new coordinates after each transformation, and the final image showcased the new seating arrangement on the coordinate plane.