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Math Help - finding cordinate of a point ( i know distance and gradient)

  1. #1
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    finding cordinate of a point ( i know distance and gradient)

    i have the the cordinate of a point

    i have the distance of a line

    i have the gradient of the line

    i want to know the cordinate of the 2nd point

    that is from the first cordinate, add the distance to it considering the gradient

    i need to know the 2nd cordinate

    i know i might get 2 cordinates but it dont matter

    how to do it can someone help?
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  2. #2
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    The gradient (slope) of a line is the tangent of the angle, \theta the line makes with the x-axis. And tan(\theta)= \frac{length of opposite side}{lenght of near side}= \frac{\Delta y}{\Delta x}. If the gradient is m then m= \frac{\Delta y}{\Delta x} or, equivalently, m\Delta x= \Delta y.

    And, of course, if length of the line is d, then d^2= \Delta x^2+ \Delta y^2 so that \Delta y^2= d^2- \Delta x^2 and \Delta y= \sqrt{d^2- \Delta x^2}.

    Putting that into the equation for the gradient, m, m\Delta x= \sqrt{d^2- \Delta x^2}.

    Square both sides to get m^2\Delta x^2= d^2- \Delta x^2 which is the same as (m^2+ 1)\Delta x^2= d^2.

    Solve that for \Delta x and then find \Delta y from \Delta y= m\Delta x

    \Delta x and \Delta y are the changes in the x and y coordinates to add then to the x and y values of the initial point to find the final point.

    Yes, there are two possible answers- the points on the same line on either side of the initial point. When you solve (m^2+ 1)\Delta x= d^2, you will need to take a square root and which point you get depends on which sign you take for \Delta x.
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