Points A and B are at #(2 ,7 )# and #(8 ,6 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #4 #. If point A is now at point B, what are the coordinates of point C?
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To find the coordinates of point C after the rotation and dilation, we can follow these steps:
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Find the new coordinates of point A after the rotation by π radians counterclockwise about the origin. This can be done using the rotation formula: New_x = Old_x * cos(θ) - Old_y * sin(θ) New_y = Old_x * sin(θ) + Old_y * cos(θ) Substituting the values, we get: New_x = 2 * cos(π) - 7 * sin(π) = -2 New_y = 2 * sin(π) + 7 * cos(π) = -7
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Next, dilate the new coordinates of point A by a factor of 4 about point C. The dilation formula is: New_x = C_x + Dilation_Factor * (Old_x - C_x) New_y = C_y + Dilation_Factor * (Old_y - C_y) Substituting the values, we get: -2 = C_x + 4 * (2 - C_x) -7 = C_y + 4 * (7 - C_y)
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Solve the system of equations to find the coordinates of point C. From the first equation: -2 = C_x + 8 - 4C_x => 2C_x = 10 => C_x = 5 From the second equation: -7 = C_y + 28 - 4C_y => 3C_y = 21 => C_y = 7
Therefore, the coordinates of point C are (5, 7).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- A line segment has endpoints at #(7 ,6 )# and #(5 ,8 )#. The line segment is dilated by a factor of #3 # around #(2 ,4 )#. What are the new endpoints and length of the line segment?
- A line segment has endpoints at #(1 ,2 )# and #(9 ,3 )#. If the line segment is rotated about the origin by #( 3 pi)/2 #, translated horizontally by # 2 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- Circle A has a radius of #3 # and a center of #(2 ,4 )#. Circle B has a radius of #2 # and a center of #(4 ,7 )#. If circle B is translated by #<2 ,-4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(7 ,6 )# and #(5 ,3 )#. If the line segment is rotated about the origin by # pi #, translated horizontally by # - 4 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- If the blueprint is drawn to scale so that every 1⁄2 inch represents 10 yds, what are the dimensions on the blueprint?

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