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Find the measure of angle from l1 to l2.

(i) l1: joining (2, 0) and (5, 0)
l2: joining (2, 0) and (5, 5)

(ii) l1: joining (-2, 1) and (3, 4)
l2: joining (-1, 3) and (4, 8)

(iii) l1: joining (-5, -4) and (5, 1)
l2: joining (-3, 2) and (0, 5)

(iv) l1: joining (2, -6) and (5, -9)
l2: joining (5, -5) and (-10, -5)

Answer :

To find the measure of the angle between two lines, we use the formula for the tangential angle between two lines given by their slopes. The formula is:

[tex]\theta = \tan^{-1} \left( \frac{m_2 - m_1}{1 + m_1 m_2} \right)[/tex]

where [tex]m_1[/tex] and [tex]m_2[/tex] are the slopes of lines [tex]l_1[/tex] and [tex]l_2[/tex] respectively.

Let's go through each of the parts:

(i) Lines joining (2, 0) and (5, 0), and (2, 0) and (5, 5):


  • Line [tex]l_1[/tex]:


    • Points: (2, 0) and (5, 0)

    • Slope [tex]m_1 = \frac{0 - 0}{5 - 2} = 0[/tex]



  • Line [tex]l_2[/tex]:


    • Points: (2, 0) and (5, 5)

    • Slope [tex]m_2 = \frac{5 - 0}{5 - 2} = \frac{5}{3}[/tex]



  • Angle calculation:


    • [tex]\theta = \tan^{-1} \left( \frac{\frac{5}{3} - 0}{1 + 0 \times \frac{5}{3}} \right) = \tan^{-1} \left( \frac{5}{3} \right)[/tex]




(ii) Lines joining (-2, 1) and (3, 4), and (-1, 3) and (4, 8):


  • Line [tex]l_1[/tex]:


    • Slope [tex]m_1 = \frac{4 - 1}{3 - (-2)} = \frac{3}{5}[/tex]



  • Line [tex]l_2[/tex]:


    • Slope [tex]m_2 = \frac{8 - 3}{4 - (-1)} = \frac{5}{5} = 1[/tex]



  • Angle calculation:


    • [tex]\theta = \tan^{-1} \left( \frac{1 - \frac{3}{5}}{1 + \frac{3}{5} \times 1} \right) = \tan^{-1} \left( \frac{2}{8} \right) = \tan^{-1} \left( \frac{1}{4} \right)[/tex]




(iii) Lines joining (-5, -4) and (5, 1), and (-3, 2) and (0, 5):


  • Line [tex]l_1[/tex]:


    • Slope [tex]m_1 = \frac{1 - (-4)}{5 - (-5)} = \frac{5}{10} = \frac{1}{2}[/tex]



  • Line [tex]l_2[/tex]:


    • Slope [tex]m_2 = \frac{5 - 2}{0 - (-3)} = \frac{3}{3} = 1[/tex]



  • Angle calculation:


    • [tex]\theta = \tan^{-1} \left( \frac{1 - \frac{1}{2}}{1 + \frac{1}{2} \times 1} \right) = \tan^{-1} \left( \frac{1}{1.5} \right) = \tan^{-1} \left( \frac{2}{3} \right)[/tex]




(iv) Lines joining (2, -6) and (5, -9), and (5, -5) and (-10, -5):


  • Line [tex]l_1[/tex]:


    • Slope [tex]m_1 = \frac{-9 - (-6)}{5 - 2} = \frac{-3}{3} = -1[/tex]



  • Line [tex]l_2[/tex]:


    • Slope [tex]m_2 = \frac{-5 - (-5)}{-10 - 5} = 0[/tex]



  • Angle calculation:


    • [tex]\theta = \tan^{-1} \left( \frac{0 - (-1)}{1 + (-1) \times 0} \right) = \tan^{-1}(1) = 45^\circ[/tex]




Thus, we've calculated the angle between each pair of lines expressed in degrees or radians based on the inverse tangent function. Always remember to ensure your calculator is set to the correct mode when calculating these angles.

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