Question

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

Answer 1

To find the other endpoint of the line segment with one endpoint at (8, -1) and a point one-third of the way at (5, -2), the changes in coordinates are calculated and projected two more times to reach the final endpoint, which is (-1, -4).

Step-by-step explanation:

The student is asking to find the other endpoint of a line segment. The known endpoint is (8, −1) and we are given a point that is one-third of the way from this endpoint to the other, which is (5, −2). Since the given point is one-third the distance towards the other endpoint, we can deduce that the segment from (8, −1) to (5, −2) should be repeated two more times to reach the other endpoint.

To find the changes in the x and y coordinates from the known endpoint to the point one-third of the way, we calculate the differences 5 - 8 = -3 (for x) and −2 - (−1) = -1 (for y). Multiplying these by 3 (since the point is a third of the way), we get -3 * 3 = -9 and -1 * 3 = -3. Now we add these results to the coordinates of the known endpoint to find the coordinates of the other endpoint: 8 + (-9) = -1 (x-coordinate) and −1 + (-3) = −4 (y-coordinate).

The other endpoint of the line segment is therefore (−1, −4).

Answer 2

The other endpoint is (-4, -5)

Step-by-step explanation:

We know the formula for the coordinates of the point dividing the line segment in the ratio of a : b is

x = x' +
`(a)/(a+b)(x`

y = y' +
`(a)/(a+b)(y`

Now we plug in the values of endpoints (8, -1) and (x, y) with a point (5, -2) dividing the segment in 1 : 3 ratio.

5 = 8 +
`(1)/(1+3)(x-8)`

5 - 8 =
`(x-8)/(4)`

(x - 8) = -3×4 = -12

x = -12 + 8 = -4

Similarly, -2 = -1 +
`(1)/(1+3)(y+1)`

-2 + 1 =
`(1)/(4)(y+1)`

y + 1 = -1(4)

y + 1 = -4

y = -1 - 4 = -5

Therefore, another endpoint of the segment is (-4, -5).