To find the coordinates of the point that divides the segment from (1,2) to (6,12) into a ratio of 3:2, we can use the following steps:
1. Find the distance between the two points using the distance formula:
d = sqrt((x2-x1)^2 + (y2-y1)^2)
d = sqrt((6-1)^2 + (12-2)^2)
d = sqrt(5^2 + 10^2)
d = sqrt(125)
d = 5sqrt(5)
2. Multiply the distance by the ratio of 3:2 to get the length of the smaller segment (3 parts) and the larger segment (2 parts):
3x = (3/5)(5sqrt(5)) = 3sqrt(5)
2x = (2/5)(5sqrt(5)) = 2sqrt(5)
3. Starting from (1,2), move a distance of 3sqrt(5) in the direction of (6,12) to find the endpoint of the smaller segment:
x = 1 + (3/5)(5)
y = 2 + (3/5)(10)
x = 4
y = 8
Endpoint of smaller segment: (4,8)
4. Starting from (1,2), move a distance of 2sqrt(5) in the direction of (6,12) to find the endpoint of the larger segment:
x = 1 + (2/5)(5)
y = 2 + (2/5)(10)
x = 3
y = 6
Endpoint of larger segment: (3,6)
Therefore, the point that divides the segment from (1,2) to (6,12) into a ratio of 3:2 is (4,8).