What are the coordinates of the point 3/4 of the way from A to B?

What are the coordinates of the point 3/4 of the way from A to B?


[tex]What are the coordinates of the point 3/4 of the way from A to B?[/tex]

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  1. (-3.5, 1.25)

    Step-by-step explanation:

    Given:

    A(-5, -4)

    B(-3, 3)

    Required:

    Coordinates of the point 3/4 of the distance between A and B.

    SOLUTION:

    Find the coordinates using the formula below:

    [tex](x, y) = (x_1 + k(x_2 - x_1), y_1 + k(y_2 - y_1))[/tex]

    Let,

    [tex]A(-5, -4) = (x_1, y_1)[/tex]

    [tex]B(-3, 3) = (x_2, y_2)[/tex]

    [tex]k = \frac{3}{4}[/tex]

    Plug in the values into the formula:

    [tex](x, y) = (-5 + \frac{3}{4}(-3 -(-5)), -4 + \frac{3}{4}(3 -(-4))[/tex]

    [tex](x, y) = (-5 + \frac{3}{4}(2), -4 + \frac{3}{4}(7)[/tex]

    [tex](x, y) = (-5 + \frac{3*2}{4}, -4 + \frac{3*7}{4}[/tex]

    [tex](x, y) = (-5 + 1.5, -4 + 5.25)[/tex]

    [tex](x, y) = (-3.5, 1.25)[/tex]

    The coordinates of the point 3/4 from A to B are (-3.5, 1.25).

  2. The coordinates of the point at 3/4 of the distance from A to B from A is (-3.5, 1.25)

    Step-by-step explanation:

    The coordinates of the point A is (-5, -4),

    The coordinates of the point B is (-3, 3)

    Let the point 3/4 from A to B = P

    The coordinates of the point 3/4 from A to B is found as follows;

    (-5 + (3/4×(-3 - (-5)), -4 + 3/4×(3 - (-4)) which gives;

    The coordinates of the point 3/4 from A to B as P(-3.5, 1.25)

    We verify the length from A to B  and from A to P as follows;

    The distance l between two points (x₁, y₁) and  (x₂, y₂) is given by the formula;

    [tex]\sqrt{\left (y_{2}-y_{1} \right )^{2}+\left (x_{2}-x_{1} \right )^{2}}[/tex]

    For AB, we have;

    [tex]Length \ of \ segment \ \overline {AB} = \sqrt{\left (3-(-4) \right )^{2}+\left ((-3)-(-5) \right )^{2}}=\sqrt{53} \approx 7.28[/tex]

    [tex]Length \ of \ segment \ \overline {AP} = \sqrt{\left (1.25-(-4) \right )^{2}+\left ((-3.5)-(-5) \right )^{2}}= \dfrac{3}{4} \cdot \sqrt{53}[/tex]

    Therefore, the point P (-3.5, 1.25) is the point 3/4 distance of A to B from A.

  3. [tex]P(x,y) = (7,6)[/tex]

    Step-by-step explanation:

    Given

    [tex]X = (1,-6)[/tex]

    [tex]Y = (9,10)[/tex]

    [tex]Point = \frac{3}{4}[/tex]

    Required

    Determine the coordinate of the point;

    First, we need to determine the ratio of the point between X and Y

    Represent the point with P

    If the distance between point X and point P is [tex]\frac{3}{4}[/tex],

    The distance between point P and point Y will be [tex]1 - \frac{3}{4} = \frac{1}{4}[/tex]

    [tex]Ratio = XP : PY[/tex]

    [tex]Ratio = \frac{3}{4} : \frac{1}{4}[/tex]

    Multiply through by 4

    [tex]Ratio = 3: 1[/tex]

    Now, the coordinate of P can be calculated using

    [tex]P(x,y) = (\frac{mx_2 + nx_1}{n+m},\frac{my_2 + ny_1}{n+m})[/tex]

    Where

    [tex]m:n = 3:1[/tex]

    [tex](x_1,y_1) = (1,-6)[/tex]

    [tex](x_2,y_2) = (9,10)[/tex]

    Substitute these values in the formula above

    [tex]P(x,y) = (\frac{3 * 9 + 1 * 1}{3+1},\frac{3 * 10 + 1 * -6}{3+1})[/tex]

    [tex]P(x,y) = (\frac{27 + 1}{4},\frac{30 -6}{4})[/tex]

    [tex]P(x,y) = (\frac{28}{4},\frac{24}{4})[/tex]

    [tex]P(x,y) = (7,6)[/tex]

  4. (3.5,5.25)

    Step-by-step explanation:

    the difference between the x values are 2 and dividing that by 4 is 0.5 and multiply by 3 as it is three quarters of the way so it equals 1.5. Move point a on its x axis to the value 3.5. the difference between the y values is 7 and dividing that by 4 is 1.75 and multiply by 3 which is 5.25. move points on its y axis to the value 1.25. the end coordinates are (3.5,5.25)

  5. Coordinates of the point P are [tex](-\frac{29}{7},-1)[/tex]

    Step-by-step explanation:

    If a point P(x, y) divides the line AB into the ratio of m : n, coordinates of this point will be,

    x = [tex]\frac{mx_2+nx_1}{m+n}[/tex]

    y = [tex]\frac{my_2+ny_1}{m+n}[/tex]

    In the given question m : n = 3 : 4

    And the coordinates of A and B are (-5, -4) and (-3, 3) respectively.

    x = [tex]\frac{3(-3)+4(-5)}{3+4}[/tex]

    x = -[tex]\frac{29}{7}[/tex]

    y = [tex]\frac{3(3)+4(-4)}{3+4}[/tex]

    y = -1

    Therefore, coordinates of the given point P are [tex](\frac{29}{7},-1)[/tex].

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