Question

Point B on a segment with endpoints A(1, −1) and C(3, 2) partitions the segment in a 1:3 ratio. Find B.

(0.5, 0.75)
(−0.25, 1.5)
(1.5, −0.25)
(0.75, 0.5)

Answer 1

To find point B that partitions segment AC in a 1:3 ratio, we can use the section formula. The formula for finding the coordinates of a point that divides a segment in the ratio \( m:n \) is given by:

\[
B = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

In this case, let \( A(1, -1) \) be point \( (x_1, y_1) \) and \( C(3, 2) \) be point \( (x_2, y_2) \). The ratio is \( m:n = 1:3 \) (meaning point B is closer to A).

So, \( m = 1 \) and \( n = 3 \).

Now substituting the coordinates and the ratio into the formula:

1. Calculate the x-coordinate of B:

\[
x_B = \frac{1 \cdot 3 + 3 \cdot 1}{1 + 3} = \frac{3 + 3}{4} = \frac{6}{4} = 1.5
\]

2. Calculate the y-coordinate of B:

\[
y_B = \frac{1 \cdot 2 + 3 \cdot (-1)}{1 + 3} = \frac{2 - 3}{4} = \frac{-1}{4} = -0.25
\]

Thus, the coordinates of point B are:

\[
B(1.5, -0.25)
\]

Out of the provided options, the correct answer is:

**(1.5, −0.25)**.