AB joins the points (-10,2) and (8,14). Find the coordinates of X and Y on AB so that AX = XY = YB...?

Answer 1

The distance between x1 and x2 (on the x-axis) = x2 - x1 = 8 - (-10)= 18
The distance between y1 and y2 (on the y-axis)= y2 - y1 = 14 - 2 = 12

You want to divide AB into 3 equal parts. So divide x=18 and y=12 by 3 and you get dx=6, dy=4

The x co-ordinate of X, would, therefore be:

x1 + dx = -10+6 = -4

and the y co-ordinate would be

y1 + dy = 2 + 4 = 6. Thus X = (-4,6)

Likewise, find the co-ordinates of Y, by adding dx and dy to the co-ordinates of X (-4,6):

Y = (2,10)

It's easy if you graph the equation.

Answer 2

This a generalization of the idea of finding the midpoint of a line segment. In finding the midpoint, you are dividing a line segment into two equal parts; here, you are asked to divide the segment into three equal parts.

Take X = (-10 +(1/3)*(8 - (-10)),2 + (1/3)*(14 - 2)) and
Y = (-10 + (2/3)*(18),2 + (2/3)*(12)).

Check that slope(AX) = slope(XY) = slope(YB) = slope(AB), and AX = XY = YB.