Question

Use the midpoint formula repeatedly to find the three points that divide the segment joining (x1, y1) and (x2, y2) into four equal parts

Answer 1

Midpoint is a point that divides a line segment into two parts that are equal or congruent. It is a point that is equidistant between the two endpoints. It would be the point that is found halfway the two endpoints. So, we obtain this point by locating the point that is in between the two endpoints. For the points (x1, y1) and (x2, y2), we determine the midpoint as follows:

(x1+x2/2 , y1+y2/2) = (x3, y3)

So, we will have three points which are (x1, y1), (x2, y2) and (x3, y3). Then, we need to use the formula again to determine the midpoint of the two line segment.


(x1+x3/2 , y1+y3/2) = (x4, y4)
Substituting the value for x3 and y3,

(x1+((x1+x2)/2)/2 , y1+((y1+y2)/2)/2) = (x4, y4)
 (x4, y4) = (3x1 + x2 / 4 , 3y1 + y2 / 4)


(x3+x2/2 , y3+y2/2) = (x5, y5)
We do the same as the procedure above,

(x2+((x1+x2)/2)/2 , y2+((y1+y2)/2)/2) = (x4, y4)
 (x5, y5) = (3x2 + x1 / 4 , 3y2 + y1 / 4)