text: # Point A is at (-1 ,-8 ) and point B is at (-5 ,3 ). Point A is rotated pi/2  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?
  
  May 21, 2016
  
  (-8 ,1) and ≈ 8.104
  
  #### Explanation:
  
  Under a rotation of $\frac{\pi}{2}$ clockwise about O.
  
  a point (x ,y) → (y ,-x)
  
  hence point A(-1 ,-8) → A' (-8 ,1)
  
  To find the change in distance between AB and A'B we require to calculate the length of AB and A'B.
  We can do this using the $\textcolor{b l u e}{\text{distance formula}}$
  
  $\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
  where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 points}$
  
  length of AB$\left({x}_{1} , {y}_{1}\right) = \left(- 1 , - 8\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 5 , 3\right)$
  
  d_(AB)=sqrt((-5+1)^2+(3+8)^2)=sqrt(16+121)≈11.705
  
  length of A'B
  $\left({x}_{1} , {y}_{1}\right) = \left(- 8 , 1\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 5 , 3\right)$
  
  d_(A'B)=sqrt((-5+8)^2+(3-1)^2)=sqrt(9+4)≈3.601
  
  hence the change in length = 11.705 - 3.601 = 8.104
explanation: The document contains a math question and its corresponding answer. The question involves the rotation of a point and the calculation of the change in distance between two points after the rotation. The answer provides the new coordinates of the rotated point and the change in distance.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Point A is at (-1 ,-8 ) and point B is at (-5 ,3 ). Point A is rotated pi/2  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?
  
  May 21, 2016
  
  (-8 ,1) and ≈ 8.104
  
  #### Explanation:
  
  Under a rotation of $\frac{\pi}{2}$ clockwise about O.
  
  a point (x ,y) → (y ,-x)
  
  hence point A(-1 ,-8) → A' (-8 ,1)
  
  To find the change in distance between AB and A'B we require to calculate the length of AB and A'B.
  We can do this using the $\textcolor{b l u e}{\text{distance formula}}$
  
  $\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
  where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 points}$
  
  length of AB$\left({x}_{1} , {y}_{1}\right) = \left(- 1 , - 8\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 5 , 3\right)$
  
  d_(AB)=sqrt((-5+1)^2+(3+8)^2)=sqrt(16+121)≈11.705
  
  length of A'B
  $\left({x}_{1} , {y}_{1}\right) = \left(- 8 , 1\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 5 , 3\right)$
  
  d_(A'B)=sqrt((-5+8)^2+(3-1)^2)=sqrt(9+4)≈3.601
  
  hence the change in length = 11.705 - 3.601 = 8.104
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding answer. The question involves the rotation of a point and the calculation of the change in distance between two points after the rotation. The answer provides the new coordinates of the rotated point and the change in distance.
  
  Label: question-and-answer