Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To determine the scale factor for the transformation of quadrilateral [tex]\( EFGH \)[/tex] to [tex]\( E'F'G'H' \)[/tex], we need to compare the coordinates of the original vertices with those of the transformed vertices. The scale factor can be determined by calculating how each coordinate in the original shape is scaled.
Given:
- Coordinates of [tex]\( EFGH \)[/tex]:
- [tex]\( E(-2, -1) \)[/tex]
- [tex]\( F(1, 2) \)[/tex]
- [tex]\( G(6, 0) \)[/tex]
- [tex]\( H(2, -2) \)[/tex]
- Coordinates of [tex]\( E'F'G'H' \)[/tex]:
- [tex]\( E'(-3, -\frac{3}{2}) \)[/tex]
- [tex]\( F'(\frac{3}{2}, 3) \)[/tex]
- [tex]\( G'(9, 0) \)[/tex]
- [tex]\( H'(3, -3) \)[/tex]
### Step-by-Step Calculation:
1. Calculate the scale factor for point [tex]\( E \)[/tex] to [tex]\( E' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{-3}{-2} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } y\text{-direction} = \frac{-\frac{3}{2}}{-1} = \frac{3}{2} \][/tex]
2. Calculate the scale factor for point [tex]\( F \)[/tex] to [tex]\( F' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{\frac{3}{2}}{1} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } y\text{-direction} = \frac{3}{2} = \frac{3}{2} \][/tex]
3. Calculate the scale factor for point [tex]\( G \)[/tex] to [tex]\( G' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{9}{6} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates ([tex]\( y \)[/tex]-coordinate of [tex]\( G \)[/tex] and [tex]\( G' \)[/tex] is 0, so it stays 0):
[tex]\[ \text{Scale factor in the } y\text{-direction} = 0 ~ (\text{undefined as direction remains 0}) \][/tex]
4. Calculate the scale factor for point [tex]\( H \)[/tex] to [tex]\( H' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{3}{2} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } y\text{-direction} = \frac{-3}{-2} = \frac{3}{2} \][/tex]
### Conclusion
We can see from above that the new coordinates are consistently scaled by a factor of [tex]\( \frac{3}{2} \)[/tex] in both the [tex]\( x \)[/tex]-direction and [tex]\( y \)[/tex]-direction (except for the [tex]\( y\)[/tex]-coordinates of [tex]\( G \)[/tex] and [tex]\( G' \)[/tex]).
Thus, the scale factor for the transformation of quadrilateral [tex]\( EFGH \)[/tex] to [tex]\( E'F'G'H' \)[/tex] is:
[tex]\[ \boxed{\frac{3}{2}} \][/tex]
Given:
- Coordinates of [tex]\( EFGH \)[/tex]:
- [tex]\( E(-2, -1) \)[/tex]
- [tex]\( F(1, 2) \)[/tex]
- [tex]\( G(6, 0) \)[/tex]
- [tex]\( H(2, -2) \)[/tex]
- Coordinates of [tex]\( E'F'G'H' \)[/tex]:
- [tex]\( E'(-3, -\frac{3}{2}) \)[/tex]
- [tex]\( F'(\frac{3}{2}, 3) \)[/tex]
- [tex]\( G'(9, 0) \)[/tex]
- [tex]\( H'(3, -3) \)[/tex]
### Step-by-Step Calculation:
1. Calculate the scale factor for point [tex]\( E \)[/tex] to [tex]\( E' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{-3}{-2} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } y\text{-direction} = \frac{-\frac{3}{2}}{-1} = \frac{3}{2} \][/tex]
2. Calculate the scale factor for point [tex]\( F \)[/tex] to [tex]\( F' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{\frac{3}{2}}{1} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } y\text{-direction} = \frac{3}{2} = \frac{3}{2} \][/tex]
3. Calculate the scale factor for point [tex]\( G \)[/tex] to [tex]\( G' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{9}{6} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates ([tex]\( y \)[/tex]-coordinate of [tex]\( G \)[/tex] and [tex]\( G' \)[/tex] is 0, so it stays 0):
[tex]\[ \text{Scale factor in the } y\text{-direction} = 0 ~ (\text{undefined as direction remains 0}) \][/tex]
4. Calculate the scale factor for point [tex]\( H \)[/tex] to [tex]\( H' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{3}{2} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } y\text{-direction} = \frac{-3}{-2} = \frac{3}{2} \][/tex]
### Conclusion
We can see from above that the new coordinates are consistently scaled by a factor of [tex]\( \frac{3}{2} \)[/tex] in both the [tex]\( x \)[/tex]-direction and [tex]\( y \)[/tex]-direction (except for the [tex]\( y\)[/tex]-coordinates of [tex]\( G \)[/tex] and [tex]\( G' \)[/tex]).
Thus, the scale factor for the transformation of quadrilateral [tex]\( EFGH \)[/tex] to [tex]\( E'F'G'H' \)[/tex] is:
[tex]\[ \boxed{\frac{3}{2}} \][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.