Coordinates question (maths)

Question

"The points A, B and C lie in order on a straight line.

 

The coordinates of A are (2, 5)

The coordinates of B are (4, p)

The coordinates of C are (q, 17)

 

Given that AC= 4AB, find the values of p and q"

 

I would like to know the working out of this as I only know the answer(p=8, q=10)

 

Thank you in advance.

Answer 1

AC = 4AB means that AC is four times the length of AB

Looking at the x-coordinate (the run), 4 - 2 = 2

We need to multiply that by 4 to see how far the run is on AC:

4 * 2 = 8

So that's how far we have the run is for AC.

Since we did not start on the y-axis, we need to add in the x-coordinate is for point A:

8 + 2 = 10, which is q

 

We can either use the two points to find the slope and from there the y-coordinate for point B, or we can use the information above again. Let's use the information given just for more practice:

AC = 4AB

 

17 - 5 = 12 (the line AC's rise goes up 12 units)

12/4 = 3 (line AB is only one-fourth as long as AC)

5 + 3 = 8 = p (Point A is already 5 units up; add 3, and we get 8 for p)