Write your question here (Keep it simple and clear to get the best answer)striangle has vertices A(2;5); B(1;-2) and C(-5;1). Determine:(a)

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Write your question here (Keep it simple and clear to get the best answer)striangle has vertices A(2;5); B(1;-2) and C(-5;1). Determine:(a) the equation of the line BC. (b) The equation of the perpendicular line from A to Bc

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Philomena 3 years 2021-08-06T04:23:49+00:00 1 Answers 6 views 0

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  1. Ladonna
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    2021-08-06T04:25:00+00:00
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    Answer:

    (a) y = \frac{-1}{2}x –  \frac{3}{2}

    (b) y = 2x + 3

    Step-by-step explanation:

    (a) The equation of a line given by points M(x₁, y₁) and N(x₂, y₂) is given by:

    y – y₁ = m(x – x₁)            ——————-(i)

    Where;

    m = \frac{y_2 - y_1}{x_2 - x_1} = slope or gradient of the line  —————(ii)

    Given points on the triangle are:

    A(2,5)

    B(1,-2)

    C(-5,1)

    To find the equation of line BC, we use the formulas in equations (i) and (ii) where the points of the line are B(1,-2) and C(-5,1) and;

    x₁ = 1

    y₁ = -2

    x₂ = -5

    y₂ = 1

    ==> First get the gradient using equation (ii) as follows;

    m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-2)}{-5 -1}

    m =   \frac{3}{-6}

    m = \frac{-1}{2}

    ==> Now, use equation (i) to find the equation of the line as follows;

    y – (-2) = \frac{-1}{2}(x – 1)

    y + 2 = \frac{-1}{2}(x – 1)

    Multiply both sides by 2

    2(y+2) = -1 ( x – 1 )

    2y + 4 = -x + 1

    2y = -x + 1 – 4

    2y = – x – 3

    y =  \frac{-1}{2}x –  \frac{3}{2}

    Therefore, the equation of the line is y = \frac{-1}{2}x –  \frac{3}{2}

    (b) To find the perpendicular line from A to BC, note that

    i. two lines are perpendicular if they meet at 90°

    ii. the general equation of a line could also be written as y = mx + c where m is the slope and c is the intercept.

    iii. when one line has a slope of m, then a perpendicular line to that line will have a slope of \frac{-1}{m}

    The equation of line BC is y = \frac{-1}{2}x –  \frac{3}{2}.

    This means that BC has a slope of \frac{-1}{2}

    A perpendicular line from A to BC will have a slope of 2.

    Now to get the equation of this perpendicular line from A(2, 5) to BC, we use the general equation of a line given in equation (i)

    where;

    m = 2

    x₁ = 2

    y₁ = 5

    Substitute these values into equation (i)

    y – 5 = 2(x – 2)

    Solving by simplification gives;

    y – 5 = 2x – 2

    y = 2x + 3

    Therefore, the equation of the perpendicular line is y = 2x + 3

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