Question
A triangle undergoes a sequence of transformations. First, the triangle is dilated by a scale factor of 1/3 about the origin. Then the triangle is reflected over the x-axis. Finally, the triangle is translated left 3 units and up 2 units.
How does the image triangle compare to the pre-image triangle?
How does the image triangle compare to the pre-image triangle?
Answers
GPT-4o mini
To analyze how the image triangle compares to the pre-image triangle after the given transformations, let's break down each transformation step by step:
1. **Dilation by a scale factor of 1/3 about the origin**: When we dilate a triangle by a scale factor of 1/3, every vertex of the triangle is moved closer to the origin. If the original vertices of the triangle are at points \((x, y)\), after dilation, the new vertices become \((\frac{1}{3}x, \frac{1}{3}y)\). This transformation reduces the size of the triangle to one-third its original size.
2. **Reflection over the x-axis**: Reflecting a shape over the x-axis changes the sign of the y-coordinates of each vertex. So if the vertices after dilation are at \((\frac{1}{3}x, \frac{1}{3}y)\), after reflection, they will be at \((\frac{1}{3}x, -\frac{1}{3}y)\). This reflection flips the triangle upside down across the x-axis.
3. **Translation left 3 units and up 2 units**: A translation is a shifting of all points of the triangle by a certain vector. Translating left 3 units means subtracting 3 from the x-coordinate, and translating up 2 units means adding 2 to the y-coordinate. Therefore, each vertex at \((\frac{1}{3}x, -\frac{1}{3}y)\) moves to the new position \((\frac{1}{3}x - 3, -\frac{1}{3}y + 2)\).
After applying all these transformations, we can summarize the final positions of the vertices. The image triangle is similar to the original triangle (because the dilation maintains the aspect ratio), but it is reduced in size to one-third of the original triangle's area.
In terms of transformations:
- The triangle is now upside down due to the reflection.
- It has been shifted left and upward relative to its position after the reflection.
In conclusion, the image triangle is a smaller version of the pre-image triangle (one-third the size, as per the dilation), and it has been reflected over the x-axis and translated.
1. **Dilation by a scale factor of 1/3 about the origin**: When we dilate a triangle by a scale factor of 1/3, every vertex of the triangle is moved closer to the origin. If the original vertices of the triangle are at points \((x, y)\), after dilation, the new vertices become \((\frac{1}{3}x, \frac{1}{3}y)\). This transformation reduces the size of the triangle to one-third its original size.
2. **Reflection over the x-axis**: Reflecting a shape over the x-axis changes the sign of the y-coordinates of each vertex. So if the vertices after dilation are at \((\frac{1}{3}x, \frac{1}{3}y)\), after reflection, they will be at \((\frac{1}{3}x, -\frac{1}{3}y)\). This reflection flips the triangle upside down across the x-axis.
3. **Translation left 3 units and up 2 units**: A translation is a shifting of all points of the triangle by a certain vector. Translating left 3 units means subtracting 3 from the x-coordinate, and translating up 2 units means adding 2 to the y-coordinate. Therefore, each vertex at \((\frac{1}{3}x, -\frac{1}{3}y)\) moves to the new position \((\frac{1}{3}x - 3, -\frac{1}{3}y + 2)\).
After applying all these transformations, we can summarize the final positions of the vertices. The image triangle is similar to the original triangle (because the dilation maintains the aspect ratio), but it is reduced in size to one-third of the original triangle's area.
In terms of transformations:
- The triangle is now upside down due to the reflection.
- It has been shifted left and upward relative to its position after the reflection.
In conclusion, the image triangle is a smaller version of the pre-image triangle (one-third the size, as per the dilation), and it has been reflected over the x-axis and translated.