Question
Line AB is on the coordinate plane and does not pass through the origin. Line AB is dilated with a scale factor of 3 and a center
Answers
GPT-4o mini
To understand how a line is dilated on a coordinate plane, we need to know two key pieces of information: the center of dilation and the scale factor. In your case, the scale factor is 3, and the line AB does not pass through the origin.
Let's assume line AB can be expressed using the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
### Steps to Dilate the Line AB
1. **Identify the Center of Dilation**: Let's say the center of dilation is the point \( C(x_c, y_c) \).
2. **Choose Points on Line AB**: Select two points on line AB. For simplicity, let those points be \( A(x_1, y_1) \) and \( B(x_2, y_2) \).
3. **Apply Dilation Formula**: The formula to find the coordinates of a dilated point \( P'(x', y') \) from a point \( P(x, y) \) with respect to a center of dilation \( C(x_c, y_c) \) is:
\[
x' = x_c + k(x - x_c)
\]
\[
y' = y_c + k(y - y_c)
\]
where \( k \) is the scale factor. For your case, \( k = 3 \).
4. **Calculate the New Points**: Apply this dilation formula to both points \( A \) and \( B \):
- For point \( A(x_1, y_1) \):
\[
x'_1 = x_c + 3(x_1 - x_c)
\]
\[
y'_1 = y_c + 3(y_1 - y_c)
\]
- For point \( B(x_2, y_2) \):
\[
x'_2 = x_c + 3(x_2 - x_c)
\]
\[
y'_2 = y_c + 3(y_2 - y_c)
\]
5. **Find the Equation of the Dilated Line**: Once you have the coordinates of the dilated points \( A' \) and \( B' \) \((x'_1, y'_1)\) and \((x'_2, y'_2)\), you can calculate the slope of the new line:
\[
m' = \frac{y'_2 - y'_1}{x'_2 - x'_1}
\]
The equation of the dilated line can then be formulated.
This process would provide you with the equation and properties of the dilated line, which will be parallel to the original line due to the nature of dilation on straight lines.
If you want to proceed with specific numerical examples or any particular points, feel free to provide those, and I can work through the calculations with you!
Let's assume line AB can be expressed using the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
### Steps to Dilate the Line AB
1. **Identify the Center of Dilation**: Let's say the center of dilation is the point \( C(x_c, y_c) \).
2. **Choose Points on Line AB**: Select two points on line AB. For simplicity, let those points be \( A(x_1, y_1) \) and \( B(x_2, y_2) \).
3. **Apply Dilation Formula**: The formula to find the coordinates of a dilated point \( P'(x', y') \) from a point \( P(x, y) \) with respect to a center of dilation \( C(x_c, y_c) \) is:
\[
x' = x_c + k(x - x_c)
\]
\[
y' = y_c + k(y - y_c)
\]
where \( k \) is the scale factor. For your case, \( k = 3 \).
4. **Calculate the New Points**: Apply this dilation formula to both points \( A \) and \( B \):
- For point \( A(x_1, y_1) \):
\[
x'_1 = x_c + 3(x_1 - x_c)
\]
\[
y'_1 = y_c + 3(y_1 - y_c)
\]
- For point \( B(x_2, y_2) \):
\[
x'_2 = x_c + 3(x_2 - x_c)
\]
\[
y'_2 = y_c + 3(y_2 - y_c)
\]
5. **Find the Equation of the Dilated Line**: Once you have the coordinates of the dilated points \( A' \) and \( B' \) \((x'_1, y'_1)\) and \((x'_2, y'_2)\), you can calculate the slope of the new line:
\[
m' = \frac{y'_2 - y'_1}{x'_2 - x'_1}
\]
The equation of the dilated line can then be formulated.
This process would provide you with the equation and properties of the dilated line, which will be parallel to the original line due to the nature of dilation on straight lines.
If you want to proceed with specific numerical examples or any particular points, feel free to provide those, and I can work through the calculations with you!