Question

What are the coordinates of the point on the directed line segment from (2,−1) to (8,7) that partitions the segment into a ratio of 1 to 3?

Answer 1

To find the coordinates of the point that partitions the directed line segment from \((2, -1)\) to \((8, 7)\) in the ratio of 1:3, we can use the section formula.

Recall the section formula, which gives us the coordinates of a point that divides a line segment internally in the ratio \(m:n\). The coordinates \((x, y)\) of the dividing point are given by:

\[
x = \frac{mx_2 + nx_1}{m+n}
\]
\[
y = \frac{my_2 + ny_1}{m+n}
\]

Here, we are given:
- \((x_1, y_1) = (2, -1)\)
- \((x_2, y_2) = (8, 7)\)
- The ratio \(m:n = 1:3\), so \(m = 1\) and \(n = 3\).

Now, we can substitute the values into the formulas:

1. Calculate the \(x\)-coordinate:

\[
x = \frac{1 \cdot 8 + 3 \cdot 2}{1 + 3} = \frac{8 + 6}{4} = \frac{14}{4} = 3.5
\]

2. Calculate the \(y\)-coordinate:

\[
y = \frac{1 \cdot 7 + 3 \cdot (-1)}{1 + 3} = \frac{7 - 3}{4} = \frac{4}{4} = 1
\]

Thus, the coordinates of the point that partitions the directed line segment in the ratio of 1:3 are:

\[
\boxed{(3.5, 1)}
\]