Math, asked by janvihatwar, 8 months ago

Find the angle between the lines joining the points
(0.0), (2, 3) and the points (2,-2), (3,5).​

Answers

Answered by SƬᏗᏒᏇᏗƦƦᎥᎧƦ
6

Solution :-

Here we've been provided with the four points in which (0 , 0) and (2 , 3) are joining each other whereas (2 , -2) and (3 , 5) are joining each other and forming two lines.

Let us denote them with A , B , C and D for better understanding.

  • A (0 , 0)
  • B (2 , 3)
  • C (2 , -2)
  • D (3 , 5)

We need to first find out slope of them so let's start !

As we know that slope (m) is calculated by the formula :

  • \red{\boxed{\bf{m \:  =  \:   \dfrac{ y_{2} \:  -  \:y_{1}}{ x_{2} \:  -  \:x_{1} } }}}

Slope of AB :

\implies\sf{m =  \:   \dfrac{3 - 0}{2 - 0 } } \\  \implies\sf{m =  \:   \dfrac{3 }{2} }

Slope of CD :

 \implies\sf{m =  \:   \dfrac{5 + 2}{3 - 2} } \\  \implies\sf{m =  \:   \dfrac{7}{1} }

Now, So as we know that when two slopes (m) of two lines are given then angle between them is calculated by the formula :

  •  \red{\boxed{\bf{tan \:  \theta \:  =  \: ± \dfrac{m_{2} -m_{1}}{1 +m_{1}. m_{2}} }}}

\implies \:  \sf{tan \:  \theta \:  =  \:  \dfrac{7 -  \dfrac{3}{2} }{ 1 + 7(\dfrac{3}{2} )} } \\ \\   \implies \:  \sf{tan \:  \theta \:  =  \:  \dfrac{7 -  \dfrac{3}{2} }{ 1 + ( \dfrac{21}{2})}} \\  \\ \implies \:  \sf{tan \:  \theta \:  =  \:  \dfrac{7 -  \dfrac{3}{2} }{ \dfrac{2 + 21}{2}}} \\  \\ \implies \:  \sf{tan \:  \theta \:  =  \:  \dfrac{7 -  \dfrac{3}{2} }{  \dfrac{23}{2}}} \\  \\ \implies \:  \sf{tan \:  \theta \:  =  \:  \dfrac{\dfrac{14 - 3}{2} }{  \dfrac{23}{2}}} \\  \\ \implies \:  \sf{tan \:  \theta \:  =  \:  \dfrac{\dfrac{11}{2} }{  \dfrac{23}{2}}} \\  \\ \implies \:  \sf{tan \:  \theta \:  =  \:  \dfrac{\dfrac{11}{ \cancel2} }{  \dfrac{23}{ \cancel2}}} \\  \\ \implies \:  \bf{tan \:  \theta \:  =  \: ± \dfrac{11}{23}}

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