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

1. ## 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?

2. 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 $\Delta x$.