Question

PLEASE SOMEBODY HELP: Generate a symbolic rule for locating the point that divides a line segment into two parts so that the ratio of the lengths is m: n, with the point closer to the left endpoint.​

Answer 1

Explanation:

Let the coordinates of the left endpoint of the line segment be (x1, y1) and the coordinates of the right endpoint be (x2, y2). Let the point dividing the line segment be (x, y). Then the distance between the left endpoint and (x, y) is mx/(m+n) and the distance between (x, y) and the right endpoint is nx/(m+n).

Using the distance formula, we have:

`distance \: between \: (x1, y1) \: and \: (x, y) = √(((x - x1)^2 + (y - y1)^2)) = (mx)/((m+n))`

`distance \: between \: (x, y) \: and \: (x2, y2) = √(((x2 - x)^2 + (y2 - y)^2)) = (nx)/((m+n))`

Squaring both equations, we get:

`(x - x1)^2 + (y - y1)^2 = (mx)/((m+n)^2)`

`(x2 - x)^2 + (y2 - y)^2 = (nx)/((m+n)^2)`

Expanding the squares, we get:

`x^2 - 2x1x + x1^2 + y^2 - 2y1y + y1^2 = (mx)/((m+n)^2)`

`x^2 - 2x2x + x2^2 + y^2 - 2y2y + y2^2 = (nx)/((m+n)^2)`

Rearranging and simplifying, we get:

`x = ((mx2 + nx1))/((m+n)) \: and \: y = ((my2 + ny1))/((m+n))`

Therefore, the point that divides the line segment into two parts so that the ratio of the lengths is m: n and the point is closer to the left endpoint is:

`(x, y) = ((mx2 + nx1))/((m+n)), ((my2 + ny1))/((m+n))`