A line segment has endpoints at #(5 ,8 )# and #(7 ,4)#. If the line segment is rotated about the origin by #pi #, translated horizontally by #-1 #, and reflected about the y-axis, what will the line segment's new endpoints be?
We can create a rule for this transformation.
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To find the new endpoints of the line segment after the transformations, follow these steps:
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Rotation about the origin by (\pi): Rotating a point ((x, y)) around the origin by (\pi) radians (180 degrees) gives the new points ((-x, -y)). Applying this to the given endpoints:
- (A(5, 8)) becomes (A'(-5, -8))
- (B(7, 4)) becomes (B'(-7, -4))
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Translation horizontally by -1: Shifting a point ((x, y)) left by 1 unit results in the new points ((x-1, y)). Applying this to the points after rotation:
- (A'(-5, -8)) becomes (A''(-6, -8))
- (B'(-7, -4)) becomes (B''(-8, -4))
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Reflection about the y-axis: Reflecting a point ((x, y)) over the y-axis gives new points ((-x, y)). Applying this to the points after translation:
- (A''(-6, -8)) becomes (A'''(6, -8))
- (B''(-8, -4)) becomes (B'''(8, -4))
Thus, the new endpoints of the line segment after all the transformations are (A'''(6, -8)) and (B'''(8, -4)).
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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.
- Circle A has a radius of #2 # and a center of #(6 ,5 )#. Circle B has a radius of #3 # and a center of #(2 ,4 )#. If circle B is translated by #<1 ,3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- Points A and B are at #(8 ,5 )# and #(2 ,3 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #3 #. If point A is now at point B, what are the coordinates of point C?
- A line segment has endpoints at #(5 ,8 )# and #(2 , 1)#. The line segment is dilated by a factor of #1/2 # around #(3 , 5)#. What are the new endpoints and length of the line segment?
- A triangle has corners at #(4, 4 )#, ( 3, -2)#, and #( 5, -3)#. If the triangle is reflected across the x-axis, what will its new centroid be?
- Point A is at #(-4 ,5 )# and point B is at #(-3 ,7 )#. Point A is rotated #pi # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?
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