# Finding the coordinates of point $R$ in a parallelogram

What i tried

I have solved Question 22(a) as shown on the image.

Its Question 22(b) that i need help in.

I have drawn the new parallelogram $AQRS$ which is the enlargement of $ABCD$ with the length of all its sides being $3$ times larger than that of parallelogram $ABCD$

Using the distance formula, I could find the length of $BA$ and multiplying this length by $3$ i could then find the length of $AQ$

Since length $AC$ has been found, multiplying this length by $3$ times i could then find length $AR$ and by Pythagoras theorem, length $QR$ can be found.

Letting point $R$ be $(x,y)$, we have from the distance formula

$$\sqrt{(x-7)^{2}+(y+12)^{2}}= |QR|$$

Im unsure of how to form the second equation in order to solve for $x$ and $y$. Could anyone explain. Thanks

To find the coordinate of $R$:
Note that $\vec{AR}=3\vec{AC}$. You have computed $\vec{AC}$ earlier, hence you should be able to compute $\vec{AR}$.
Note that $\vec{OR}=\vec{OA}+\vec{AR}$, hence you should be able to compute coordinate of $R$.
If you insist on finding another equation to solve, you can compute the equation of straight line $AC$. We know the direction and a point on it, so we can find such an equation. Solving it will give you two solutions of which you have to pick the one that going in the right direction. I don't think such approach is efficient as compared to directly using vectors.