Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine the rule describing the transformation, we need to analyze how each point [tex]\( (x, y) \)[/tex] is transformed into its new position [tex]\( (x', y') \)[/tex]. Specifically, since we are looking at dilation, we need to find the dilation factor.
Given points:
[tex]\[ E(-2,-1), F(-1,1), G(2,0) \][/tex]
Transformed points:
[tex]\[ E'(-5,-2.5), F'(-2.5,2.5), G'(5,0) \][/tex]
### Step-by-Step Solution:
#### Step 1: Understand the Concept of Dilation
Dilation is a transformation that produces an image that is the same shape as the original, but is resized by a scale factor. The scale factor, [tex]\( k \)[/tex], is the ratio of a coordinate of the image to the corresponding coordinate of the pre-image.
#### Step 2: Calculate the Scale Factor for Each Point
1. Calculate the scale factor using point [tex]\( E \)[/tex] and [tex]\( E' \)[/tex]:
[tex]\[ k_E = \frac{E_x'}{E_x} = \frac{-5}{-2} = 2.5 \quad \text{and} \quad k_E = \frac{E_y'}{E_y} = \frac{-2.5}{-1} = 2.5 \][/tex]
Both [tex]\( k_E \)[/tex] values are equal to 2.5.
2. Calculate the scale factor using point [tex]\( F \)[/tex] and [tex]\( F' \)[/tex]:
[tex]\[ k_F = \frac{F_x'}{F_x} = \frac{-2.5}{-1} = 2.5 \quad \text{and} \quad k_F = \frac{F_y'}{F_y} = \frac{2.5}{1} = 2.5 \][/tex]
Both [tex]\( k_F \)[/tex] values are equal to 2.5.
3. Calculate the scale factor using point [tex]\( G \)[/tex] and [tex]\( G' \)[/tex]:
[tex]\[ k_G = \frac{G_x'}{G_x} = \frac{5}{2} = 2.5 \quad \text{and} \quad k_G = \frac{G_y'}{G_y} = \frac{0}{0} \][/tex]
Since [tex]\( G_y = 0 \)[/tex] and [tex]\( G_y' = 0 \)[/tex], this part is not useful for calculating the scale factor, but [tex]\( k_G \)[/tex] for [tex]\( x \)[/tex]-coordinates confirms that it is 2.5.
#### Step 3: Conclusion
Since all computed dilation factors for the respective points are consistent, we conclude that the dilation factor is [tex]\( 2.5 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \text{Dilation of } 2.5 \][/tex]
Given points:
[tex]\[ E(-2,-1), F(-1,1), G(2,0) \][/tex]
Transformed points:
[tex]\[ E'(-5,-2.5), F'(-2.5,2.5), G'(5,0) \][/tex]
### Step-by-Step Solution:
#### Step 1: Understand the Concept of Dilation
Dilation is a transformation that produces an image that is the same shape as the original, but is resized by a scale factor. The scale factor, [tex]\( k \)[/tex], is the ratio of a coordinate of the image to the corresponding coordinate of the pre-image.
#### Step 2: Calculate the Scale Factor for Each Point
1. Calculate the scale factor using point [tex]\( E \)[/tex] and [tex]\( E' \)[/tex]:
[tex]\[ k_E = \frac{E_x'}{E_x} = \frac{-5}{-2} = 2.5 \quad \text{and} \quad k_E = \frac{E_y'}{E_y} = \frac{-2.5}{-1} = 2.5 \][/tex]
Both [tex]\( k_E \)[/tex] values are equal to 2.5.
2. Calculate the scale factor using point [tex]\( F \)[/tex] and [tex]\( F' \)[/tex]:
[tex]\[ k_F = \frac{F_x'}{F_x} = \frac{-2.5}{-1} = 2.5 \quad \text{and} \quad k_F = \frac{F_y'}{F_y} = \frac{2.5}{1} = 2.5 \][/tex]
Both [tex]\( k_F \)[/tex] values are equal to 2.5.
3. Calculate the scale factor using point [tex]\( G \)[/tex] and [tex]\( G' \)[/tex]:
[tex]\[ k_G = \frac{G_x'}{G_x} = \frac{5}{2} = 2.5 \quad \text{and} \quad k_G = \frac{G_y'}{G_y} = \frac{0}{0} \][/tex]
Since [tex]\( G_y = 0 \)[/tex] and [tex]\( G_y' = 0 \)[/tex], this part is not useful for calculating the scale factor, but [tex]\( k_G \)[/tex] for [tex]\( x \)[/tex]-coordinates confirms that it is 2.5.
#### Step 3: Conclusion
Since all computed dilation factors for the respective points are consistent, we conclude that the dilation factor is [tex]\( 2.5 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \text{Dilation of } 2.5 \][/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
