A line segment has endpoints at #(3 , 1)# and #(2 ,3)#. If the line segment is rotated about the origin by #(3pi)/2 #, translated horizontally by #7#, and reflected about the x-axis, what will the line segment's new endpoints be?

Answer 1

#(8,3)" and " (10,2)#

#"label the endpoints " A(3,1)" and " B(2,3)#
#color(blue)"First transformation "" under a rotation about O of " (3pi)/2#
#"a point " (x,y)to(y,-x)#
#rArrA(3,1)toA'(1,-3)" and " B(2,3)toB'(3,-2)#
#color(blue)"Second transformation"" under a translation " ((7),(0))#
#"a point " (x,y)to(x+7,y)" hence"#
#A'(1,-3)toA''(8,-3),B'(3,-2)toB''(10,-2)#
#color(blue)"Third transformation"" reflection in x-axis"#
#"a point " (x,y)to(x,-y)" hence"#
#A''(8,-3)toA'''(8,3),B''(10,-2)toB'''(10,2)#
#"After all 3 transformations"#
#(3,1)to(8,3)" and " (2,3)to(10,2)#
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Answer 2

To find the new endpoints of the line segment after the described transformations:

  1. Rotate about the origin by ( \frac{3\pi}{2} ) radians: The rotation of a point ( (x, y) ) about the origin by ( \theta ) radians is given by: [ (x', y') = (x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta)) ] Applying this formula to both endpoints, we get: [ (3, 1) \rightarrow (1, -3) \quad \text{and} \quad (2, 3) \rightarrow (-3, 2) ]

  2. Translate horizontally by 7: Adding 7 to the x-coordinates of both endpoints, we get: [ (1, -3) \rightarrow (8, -3) \quad \text{and} \quad (-3, 2) \rightarrow (4, 2) ]

  3. Reflect about the x-axis: The reflection of a point ( (x, y) ) about the x-axis is given by ( (x, -y) ). Applying this to both endpoints, we get: [ (8, -3) \rightarrow (8, 3) \quad \text{and} \quad (4, 2) \rightarrow (4, -2) ]

Therefore, the new endpoints of the line segment are ( (8, 3) ) and ( (4, -2) ).

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Answer from HIX Tutor

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.

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