A line segment with endpoints at #(1 , 4 )# and #(2, 7 )# is rotated clockwise by #(3 pi)/2#. What are the new endpoints of the line segment?
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To find the new endpoints after rotating the line segment clockwise by ( \frac{3\pi}{2} ), you can use the following steps:

Find the midpoint of the original line segment using the formula: [ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Rotate the midpoint using the rotation formula for a point ( (x, y) ) about the origin by angle ( \theta ): [ \left( x' = x \cos(\theta)  y \sin(\theta), \quad y' = x \sin(\theta) + y \cos(\theta) \right) ]

Add or subtract the distances between the original endpoints and the midpoint to find the new endpoints.
Using these steps:

Midpoint: [ \left( \frac{1 + 2}{2}, \frac{4 + 7}{2} \right) = \left( \frac{3}{2}, \frac{3}{2} \right) ]

Rotating the midpoint by ( \frac{3\pi}{2} ): [ \left( x' = \frac{3}{2} \cos\left(\frac{3\pi}{2}\right)  \frac{3}{2} \sin\left(\frac{3\pi}{2}\right), \quad y' = \frac{3}{2} \sin\left(\frac{3\pi}{2}\right) + \frac{3}{2} \cos\left(\frac{3\pi}{2}\right) \right) ]

Calculating: [ x' = \frac{3}{2} \times 0  \frac{3}{2} \times (1) = \frac{3}{2} ] [ y' = \frac{3}{2} \times (1) + \frac{3}{2} \times 0 = \frac{3}{2} ]

The new midpoint is: ( \left( \frac{3}{2}, \frac{3}{2} \right) )

Using the new midpoint to find the new endpoints: [ \text{Endpoint 1}: \left( 1  \frac{3}{2}, 4  \left(\frac{3}{2}\right) \right) = \left( \frac{1}{2}, \frac{5}{2} \right) ] [ \text{Endpoint 2}: \left( 2  \frac{3}{2}, 7  \left(\frac{3}{2}\right) \right) = \left( \frac{1}{2}, \frac{17}{2} \right) ]
So, the new endpoints of the line segment after rotating it clockwise by ( \frac{3\pi}{2} ) are ( \left( \frac{1}{2}, \frac{5}{2} \right) ) and ( \left( \frac{1}{2}, \frac{17}{2} \right) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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