Question

question 7 (multiple choice worth 1 points) (04.03 mc) point b on a segment with endpoints a(2, -1) and c(4, 2) partitions the segment in a 1:3 ratio. find b.
(0.5, 0.75)
(-0.25, 2.5)
(0.75, 0.5)
(2.5, -0.25)
question 8 (multiple choice worth 1 points)

Explanation:

Answer:

D. (2.5, -0.25)

To find the coordinates of point \( B \) that partitions the segment \( AC \) with endpoints \( A(2, -1) \) and \( C(4, 2) \) in the ratio \( 1:3 \), we use the section formula. The section formula for a point \( (x, y) \) that divides the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) is given by:

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

Here, \( m = 1 \), \( n = 3 \), \( (x_1, y_1) = (2, -1) \), and \( (x_2, y_2) = (4, 2) \).

Step 1: Calculate the x-coordinate of \( B \)

\[
x = \frac{1 \cdot 4 + 3 \cdot 2}{1 + 3} = \frac{4 + 6}{4} = \frac{10}{4} = 2.5
\]

Step 2: Calculate the y-coordinate of \( B \)

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

Thus, the coordinates of point \( B \) are \( (2.5, -0.25) \), which corresponds to option D.