Mathematics

Question

Given points A (1, 2/3), B (x, -4/5), and C (-1/2, 4) determine the value of x such that all three points are collinear

1 Answer

  • Answer:

    [tex]x=\frac{83}{50}[/tex]

    Step-by-step explanation:

    we know that

    If the three points are collinear

    then

    [tex]m_A_B=m_A_C[/tex]

    we have

    A (1, 2/3), B (x, -4/5), and C (-1/2, 4)

    The formula to calculate the slope between two points is equal to

    [tex]m=\frac{y2-y1}{x2-x1}[/tex]

    step 1

    Find the slope AB

    we have

    [tex]A(1,\frac{2}{3}),B(x,-\frac{4}{5})[/tex]

    substitute in the formula

    [tex]m_A_B=\frac{-\frac{4}{5}-\frac{2}{3}}{x-1}[/tex]

    [tex]m_A_B=\frac{\frac{-12-10}{15}}{x-1}[/tex]

    [tex]m_A_B=-\frac{22}{15(x-1)}[/tex]

    step 2

    Find the slope AC

    we have

    [tex]A(1,\frac{2}{3}),C(-\frac{1}{2},4)[/tex]

    substitute in the formula

    [tex]m_A_C=\frac{4-\frac{2}{3}}{-\frac{1}{2}-1}[/tex]

    [tex]m_A_C=\frac{\frac{10}{3}}{-\frac{3}{2}}[/tex]

    [tex]m_A_C=-\frac{20}{9}[/tex]

    step 3

    Equate the slopes

    [tex]m_A_B=m_A_C[/tex]

    [tex]-\frac{22}{15(x-1)}=-\frac{20}{9}[/tex]

    solve for x

    [tex]15(x-1)20=22(9)[/tex]

    [tex]300x-300=198[/tex]

    [tex]300x=198+300[/tex]

    [tex]300x=498[/tex]

    [tex]x=\frac{498}{300}[/tex]

    simplify

    [tex]x=\frac{83}{50}[/tex]

NEWS TODAY