What are the coordinates of the point on the directed line segment from (−6,−4) to (3,−1) that partitions the segment into a ratio of 1 to 2?
To find the coordinates of the point that divides the line segment into a ratio of 1 to 2, we can use the formula:
(x, y) = (x1 + (1/3) * (x2 - x1), y1 + (1/3) * (y2 - y1))
Given the coordinates of the endpoints:
(x1, y1) = (-6, -4)
(x2, y2) = (3, -1)
We can substitute these values into the formula:
(x, y) = (-6 + (1/3) * (3 - (-6)), -4 + (1/3) * (-1 - (-4)))
Simplifying the equation:
(x, y) = (-6 + (1/3) * 9, -4 + (1/3) * 3)
(x, y) = (-6 + 3, -4 + 1)
(x, y) = (-3, -3)
Therefore, the coordinates of the point on the line segment that divides it in a ratio of 1 to 2 are (-3, -3).
To find the coordinates of the point that partitions the line segment into a ratio of 1 to 2, you can use the concept of the section formula.
The section formula states that if we have two points A(x₁, y₁) and B(x₂, y₂), and we divide the line segment AB into a ratio of m:n (m + n ≠ 0), the coordinates of the dividing point P are given by:
P(x,y) = ((mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n))
In this case, we have two points: A(-6,-4) and B(3,-1). We want to divide the line segment AB into a ratio of 1:2, which means m = 1 and n = 2.
Let's substitute these values into the formula:
P(x,y) = ((1 * 3 + 2 * -6) / (1 + 2), (1 * -1 + 2 * -4) / (1 + 2))
Simplifying this expression gives us:
P(x,y) = (-9 / 3, -9 / 3)
Therefore, the coordinates of the point that partitions the line segment into a ratio of 1 to 2 are:
P(-3, -3)
The ratio of 1 to 2 means that the length of the line segment from the starting point to the partition point is one-third of the total length of the segment.
Using the distance formula, we find that the length of the line segment from (-6,-4) to (3,-1) is $\sqrt{(-6-3)^2 + (-4-(-1))^2} = \sqrt{81+9} = \sqrt{90} = 3\sqrt{10}.$ The partition point is located $\frac{1}{3}$ of this distance away from the starting point.
Therefore, the coordinates of the partition point can be found by finding $\frac{1}{3}$ of the difference between the coordinates of the two points.
The x-coordinate of the partition point is $\frac{1}{3} (3 - (-6)) = \frac{1}{3} (9) = 3.$
The y-coordinate of the partition point is $\frac{1}{3}((-1) - (-4)) = \frac{1}{3}(3) = 1.$
Therefore, the coordinates of the partition point are $\boxed{(3,1)}.$
[asy]
unitsize(0.5 cm);
draw((-7,0)--(4,0));
draw((0,-5)--(0,2));
draw((-6,-4)--(3,-1),red);
draw((0,0)--(3,0),Arrow(6));
draw((0,0)--(0,1),Arrow(6));
dot("$(3,1)$",(3,1),NE);
dot("$(3,0)$",(3,0),SE);
[/asy]