coordinate geometry!

[soopertaco]

could anyone please help me with question 3e in the extended response of the SAC attached?

I can only think of a vector method but since this is methods i don't think my teacher would be 100% happy with it!

thanks in advance

[brightsky]

Plot the points. Then make a right angled triangle (dropping point B and extending point A to the right). Now cut the line AB into thirds, putting a dot at every third. The point we want is the dot nearest to B (i.e. two dots from A).

Look at the base of the triangle. It is 2 units long. The x coordinates of each of the dots marking a third would be 2/3 away from each other. Now because our one is two dots from A, then it is -8 + 2*2/3 = -8 + 4/3 = -20/3. Do the same thing for the y-coordinates. The height of our little triangle is 8 units. Hence every dot is 8/3 away from each other. Our one is the second away from A, so it is 2 + 2*8/3 = 22/3. So the point P is (-20/3, 22/3).

Notice that all this is okay in maths because of similar triangle properties.