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    Equations of Lines and Intersections

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    The diagram below (see attachment) shows a triangle ABC whose vertices are at A (-1, 3), B (6, 5) and C (8, -3). The line BP is perpendicular to the line AC, and M is the midpoint of BC.

    Note that BP is called an altitude of triangle ABC and that AM is called a median of triangle ABC.

    a) Find the gradient of
    i) The line AC
    ii) The altitude BP
    (Show answers as a fraction in its simplest form)

    b) Find the equation of the altitude BP, expressing your answer in the form: y = mx + c

    c) Find the coordinates of M.

    d) Find the equation of the median AM, expressing your answer in the form: y = mx + c

    e) Explain how you would find the coordinates of the point of intersection of the altitude BP and the median AM.

    © BrainMass Inc. brainmass.com December 24, 2021, 5:13 pm ad1c9bdddf
    https://brainmass.com/math/triangles/equations-lines-intersections-35782

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    SOLUTION This solution is FREE courtesy of BrainMass!

    a) Find the gradient of
    gradient of a line (also called slope ) = (y2-y1) /(x2-x1)

    i) The line AC
    Gradient of a line (also called slope ) = (y2-y1) /(x2-x1)

    A (-1,3) , C (8,-3)

    Therefore gradient ={ (-3)-(3) }/{ (8)-(-1)} = -6/9 = -2/3

    Answer: slope of line AC= -2/3
    ii) The altitude BP

    Altitude BP is perpendicular to AC
    The product of the gradients of perpendicular lines = -1
    Therefore ,
    Slope of line AC x slope of line BP = -1
    Or ( -2/3) x slope of line BP = -1
    Or slope of line BP = -1 / (-2/3) = 3 /2

    Answer: slope of line BP= 3/2

    (Show answers as a fraction in its simplest form)

    b) Find the equation of the altitude BP, expressing your answer in the form
    y = mx + c

    y = mx + c
    But we have calculated m = 3/2 in part a above
    Thus
    y = 3/2 x + c
    This line passes through B whose coordinates are (6,5)
    Thus the coordinates of B must satisfy the equation of line BP

    Therefore
    5 = 3/2 (6 ) + c
    or 5= 9 + c
    or c= -4

    Therefore the equation of line BP is y= 3/2 x - 4

    Answer: Equation of altitude BP is y= 3/2 x - 4

    c) Find the coordinates of M.

    The coordinates of a point which divides a straight line internally in the ratio m1:m2 is given by
    x coordinate = (m1x1 + m2x2 ) / (m1+ m2)
    y coordinate = (m1y1 + m2y2 ) / (m1+ m2)
    Here m1=m2=1 (M is the midpoint of BC. )
    x1= 6 , y1=5
    x2= 8 , y2=-3

    Therefore coordinates of M are
    x coordinate = (6 + 8 ) / (2)= 7
    y coordinate = {5+ ( -3 )} / (2)= 1

    Thus the coordinates of M is (7,1)

    d) Find the equation of the median AM, expressing your answer in the form
    y = mx + c

    The coordinates of
    A( -1,3)
    M (7,1)
    Therefore slope= (1-3) / { ( 7 ) - (-1) } = -2 / 8 = - ¼

    Thus the equation of the line AM is y = - ¼ x + c
    This line passes through A( -1,3)
    Therefore
    3 = -(1/4 ) x (-1) + c
    or c = 3 - ¼ = 11/4

    or Equation of line AM is
    y= - ¼ x + 11/4

    Answer: Equation of line AM is y= - ¼ x + 11/4

    e) Explain how you would find the coordinates of the point of intersection of the altitude BP and the median AM.

    We have two equations
    Equation of line BP is y= 3/2 x - 4
    Equation of line AM is y= - ¼ x + 11/4

    We will solve these equations simultaneously to find the coordinates of the point of intersection of the altitude BP and the median AM.

    If we do so we get
    3/2 x - 4 = - ¼ x + 11/4
    or 7 / 4 x = 27 /4
    or x= 27/7

    Substituting the value of x in y= 3/2 x - 4
    y= 25/14
    Therefore the coordinates of the point of intersection of the altitude BP and the median AM are ( 27/7 , 25/14)

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 5:13 pm ad1c9bdddf>
    https://brainmass.com/math/triangles/equations-lines-intersections-35782

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