To find the other end of the line segment, let's denote the unknown endpoint as (x, y). The given endpoint is (5, -2), and the dividing point is (8, 4), which divides the segment in the ratio 3:1.
We use the section formula for internal division. If a line segment with endpoints (x1, y1) and (x2, y2) is divided by a point (x, y) in the ratio m:n, then the coordinates of the dividing point are given by:
x = (mx2 + nx1) / (m + n)
y = (my2 + ny1) / (m + n)
Here, (x1, y1) is the unknown endpoint (x, y) that we want to find, and (x2, y2) is the known endpoint (5, -2). The dividing point (x, y) is given as (8, 4), and the ratio of division m:n is 3:1.
Plugging the given values into the section formula, we get:
8 = (3 5 + 1 x) / (3 + 1)
4 = (3 (-2) + 1 y) / (3 + 1)
Now we solve for x and y:
8 = (15 + x) / 4 ➔ 32 = 15 + x ➔ x = 32 - 15 ➔ x = 17
4 = (-6 + y) / 4 ➔ 16 = -6 + y ➔ y = 16 + 6 ➔ y = 22
Therefore, the other endpoint of the line segment is (17, 22).
