Question

Find the co-ordinates of the points dividing the line segment joining the points (−3, 5) and (5, 1) into four equal parts.

Answer 1

If P divides AB in the ratio m : n,

Then P = `((mx_2 + nx_1) / (m + n), (my_2 + ny_1) / (m + n))`

A point t of the way from A to B is (1 − t)x₁ + tx₂, (1 − t)y₁ + ty₂

Given A = (−3, 5) and B = (5, 1)

(B − A) = (5 − (−3), 1 − 5)

= (8, −4)

One quarter of vector = `(8 / 4, −4 / 4)` = (2, −1)

So, each equal step along the segment is added (2, −1) to the previous point.

Here, start at A and add the quarter-step:

⇒ First point:

= A + (2, −1)

= (−3 + 2, 5 − 1)

= (1, −4)

⇒ Second point:

= A + 2(2, −1)

= (−3 + 4, 5 − 2)

= (1, 3)

⇒ Third point:

= A + 3(2, −1)

= (−3 + 6, 5 − 3)

= (3, 2)

Hence, the final answer from A toward B is (−1, 4), (1, 3), (3, 2).