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.
can someone explain how u get -4,-1
so if
A = (8,-1)
B = (5,-2)
The distance from A to B is (-3,-1)
That means that the other endpoint is
A + 3(-3,-1) = (8,-1) + (-9,-3) = (-1,-4)
To find the other endpoint, we can use the concept of dividing a line segment into a specific ratio.
Let's say the coordinates of the other endpoint are (x, y).
According to the problem, the point (5, -2) is one-third of the way from the given endpoint (8, -1) to the other endpoint (x, y).
We can use the formula for finding a point that divides a line segment in a specific ratio:
(x2, y2) = ((1 - r) * x1 + r * x3, (1 - r) * y1 + r * y3)
where (x1, y1) and (x3, y3) are the endpoints of the line segment, and r is the ratio at which the line segment is divided.
In this case, let's assume that (8, -1) is (x1, y1), (5, -2) is (5, -2), and (x, y) is (x3, y3). Since the point (5, -2) is one-third of the way, the ratio r is equal to 1/3.
Plugging the values into the formula, we get:
(x, y) = ((1 - 1/3) * 8 + (1/3) * 5, (1 - 1/3) * -1 + (1/3) * -2)
= ((2/3) * 8 + (1/3) * 5, (2/3) * -1 + (1/3) * -2)
= (16/3 + 5/3, -2/3 - 2/3)
= (21/3, -4/3)
= (7, -4/3)
Since the coordinates are typically expressed as whole numbers, we can multiply both x and y coordinates by 3 to eliminate the fraction:
(x, y) = (7 * 3, -4/3 * 3)
= (21, -4)
Therefore, the other endpoint is (21, -4).
To find the other endpoint, we can use the concept of dividing a line segment in a given ratio. In this case, we are told that the point (5, -2) is one-third of the way from the given endpoint (8, -1) to the other endpoint.
To find the other endpoint, we use the formula for dividing a line segment in a given ratio. The formula is:
(x, y) = ((1 - t) * (x1, y1) + t * (x2, y2))
Where (x1, y1) and (x2, y2) are the coordinates of the given endpoints, t is the ratio, and (x, y) are the coordinates of the point we want to find.
In this case:
(x1, y1) = (8, -1) (given endpoint)
(x2, y2) = (x, y) (other endpoint we want to find)
t = 1/3 (since the point (5, -2) is one-third of the way)
Substituting the values into the formula, we get:
(x, y) = ((1 - 1/3) * (8, -1) + 1/3 * (5, -2))
= ((2/3) * (8, -1) + 1/3 * (5, -2))
Now, calculate each term of the equation:
(2/3) * (8, -1) = (2/3 * 8, 2/3 * -1)
= (16/3, -2/3)
(1/3) * (5, -2) = (1/3 * 5, 1/3 * -2)
= (5/3, -2/3)
Adding the two results together:
(16/3, -2/3) + (5/3, -2/3) = (16/3 + 5/3, -2/3 - 2/3)
= (21/3, -4/3)
= (7, -4/3)
Simplifying, we get the other endpoint as (7, -4/3).
Converting the fraction -4/3 to a decimal gives -1.33. Therefore, the other endpoint is (7, -1.33) or approximated as (7, -1).