for the line segment whose endpoints are r(1,2) and s(6,7), find the point located 3/4 the distance from r to s.
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To find the point located 3/4 the distance from point R (1,2) to point S (6,7), we can follow these steps:
Step 1: Find the distance between points R and S using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Given:
Point R: (1,2)
Point S: (6,7)
Substituting the values into the formula:
Distance = sqrt((6 - 1)^2 + (7 - 2)^2)
Distance = sqrt(5^2 + 5^2)
Distance = sqrt(25 + 25)
Distance = sqrt(50)
Distance = 5sqrt(2)
Step 2: Find the distance 3/4 of the total distance between R and S:
Distance 3/4 = (3/4) * (5sqrt(2))
Simplifying:
Distance 3/4 = 15/4 * sqrt(2)
Distance 3/4 = (15/4) * sqrt(2)
Distance 3/4 = 15sqrt(2)/4
Distance 3/4 = 15sqrt(2)/4
Step 3: Find the coordinates of the point located 3/4 of the distance from R to S:
To get the coordinates, we need to find the x and y coordinates separately.
For the x-coordinate:
The difference in x-coordinates between R and S is 6 - 1 = 5.
Since we are looking for a point located 3/4 of the distance, we need to find 3/4 the difference in x-coordinates:
x-coordinate = 1 + (3/4) * 5
x-coordinate = 1 + (3/4) * 5
x-coordinate = 1 + (15/4)
x-coordinate = 19/4
For the y-coordinate:
The difference in y-coordinates between R and S is 7 - 2 = 5.
Since we are looking for a point located 3/4 of the distance, we need to find 3/4 the difference in y-coordinates:
y-coordinate = 2 + (3/4) * 5
y-coordinate = 2 + (3/4) * 5
y-coordinate = 2 + (15/4)
y-coordinate = 23/4
Therefore, the point located 3/4 of the distance from R to S is (19/4, 23/4).
To find the point located 3/4 of the distance from point r(1, 2) to point s(6, 7), we need to first calculate the distance between the two points and then find the coordinates of the point that is 3/4 of that distance away from point r.
Step 1: Calculate the distance between points r and s.
We can use the distance formula to find the distance between two points in a plane (x1, y1) and (x2, y2):
d = √[(x2 - x1)^2 + (y2 - y1)^2]
Using the coordinates of points r(1,2) and s(6,7), we can substitute the values into the distance formula:
d = √[(6 - 1)^2 + (7 - 2)^2]
= √[5^2 + 5^2]
= √[25 + 25]
= √50
So, the distance between points r and s is √50.
Step 2: Find the coordinates of the point located 3/4 of the distance away from point r.
To find the coordinates, we consider the x and y direction separately. We move 3/4 of the total distance from the starting x-coordinate (x1 = 1) to the ending x-coordinate (x2 = 6), and we do the same for the y-coordinate.
For the x-coordinate:
x = x1 + (3/4) * (x2 - x1)
= 1 + (3/4) * (6 - 1)
= 1 + (3/4) * 5
= 1 + 15/4
= 1 + 3.75
= 4.75
For the y-coordinate:
y = y1 + (3/4) * (y2 - y1)
= 2 + (3/4) * (7 - 2)
= 2 + (3/4) * 5
= 2 + 15/4
= 2 + 3.75
= 5.75
Therefore, the coordinates of the point located 3/4 of the distance from r to s are (4.75, 5.75).
r + .75(s-r)
Just subtract r from s and add 3/4 of that to r.