One endpoint of a line segment is (8, -1). The point (5, -2) is one-third of the way from that endpoint to the other endpoint. Find the other endpoint.
(-1,-4)
To find the other endpoint, we need to first understand what it means for a point to be one-third of the way from one endpoint to the other endpoint.
Let's denote the coordinates of the first endpoint as (x1, y1) and the other endpoint as (x2, y2). According to the problem, we know that one endpoint is (8, -1), let's assume this is the first endpoint, so x1 = 8 and y1 = -1.
Now, let's use the midpoint formula to find the midpoint between the two endpoints. The midpoint is the point that lies exactly halfway between two given points, and it can be calculated using the following formulas:
x = (x1 + x2) / 2
y = (y1 + y2) / 2
Since we know one of the endpoints is (8, -1) and one-third of the way from this endpoint is (5, -2), we can set up the following equation using the midpoint formula:
(5, -2) = ((8 + x2) / 2, (-1 + y2) / 2)
Now, we solve for x2 and y2 in this equation:
(5, -2) = ((8 + x2) / 2, (-1 + y2) / 2)
To eliminate the fractions, you can multiply both sides of the equation by 2:
2 * (5, -2) = (8 + x2, -1 + y2)
Simplifying this equation, we get:
(10, -4) = (8 + x2, -1 + y2)
Now, we can separate the equation into two parts, one for the x-coordinate and one for the y-coordinate:
10 = 8 + x2 -- subtract 8 from both sides of the equation
-4 = -1 + y2 -- add 1 to both sides of the equation
Simplifying further:
x2 = 10 - 8 = 2
y2 = -4 + 1 = -3
Therefore, the other endpoint must be (2, -3).