Hello scubasteve94 Originally Posted by
scubasteve94 If A = (-4, 6) and B = (6, -7) find:
I) the coordinates of P, where P € AB and AP:PB = 3:1
ii) the coordinates of P, where P € AB and AP:AB = 3:1
I'm sure you know the formula for finding the coordinates of the mid-point of the line joining $\displaystyle A\;(x_1,y_1)$ to $\displaystyle B\;(x_2,y_2)$. It is:$\displaystyle \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
Well, there's a more general formula for the coordinates of the point $\displaystyle P$ that divides $\displaystyle AB$ in the ratio $\displaystyle m:n$. It is this:$\displaystyle \left(\frac{nx_1+mx_2}{m+n},\frac{ny_1+my_2}{m+n}\ right)$
(If you understand vector notation, this is equivalent to $\displaystyle \textbf{p}=\frac{n\textbf{a}+m\textbf{b}}{m+n}$, where $\displaystyle \textbf{a},\textbf{b},\textbf{p}$ are the position vectors of $\displaystyle A, B$ and $\displaystyle P$.)
For part (i), $\displaystyle AP:PB = 3:1$, so $\displaystyle m=3, n = 1$. So plug these values into the formula, using $\displaystyle x_1=-4,y_1=6,x_2=6,y_2=-7$.
For part (ii), we have $\displaystyle AP:AB=3:1$. So $\displaystyle P$ lies outside the line segment $\displaystyle AB$. Draw a diagram if you're not sure. You'll then find that the ratio $\displaystyle AP:PB = 3:-2$. So you'll need to use the values $\displaystyle m = 3, n = -2$.
Here are my answers:
Can you complete it now?
Grandad