Question

There's a segment line that goes from (2, 5) to (10, 5). Where should a point be placed on the segment to split it into a ratio 1:3 between the left and right parts?

Answer 1

To find the point that splits the segment line from (2, 5) to (10, 5) in a ratio of 1:3, we can use the concept of section formulas.

Given points \( A(2, 5) \) and \( B(10, 5) \), the ratio is 1:3. This means that the segment is divided into two parts where the left part (from A to the dividing point) is 1 unit and the right part (from the dividing point to B) is 3 units.

Let the dividing point be \( P(x, y) \). The coordinates of point \( P \) that divides the segment \( AB \) in the ratio \( m:n \) can be calculated using the following formulas:

\[
x = \frac{n \cdot x_1 + m \cdot x_2}{m+n}
\]
\[
y = \frac{n \cdot y_1 + m \cdot y_2}{m+n}
\]

In this case, \( m = 1 \), \( n = 3 \), \( x_1 = 2 \), \( y_1 = 5 \), \( x_2 = 10 \), and \( y_2 = 5 \).

Let's calculate the x-coordinate of \( P \):

\[
x = \frac{3 \cdot 2 + 1 \cdot 10}{1+3} = \frac{6 + 10}{4} = \frac{16}{4} = 4
\]

Now, since both points A and B have the same y-coordinate, the y-coordinate of point \( P \) will also be the same:

\[
y = \frac{3 \cdot 5 + 1 \cdot 5}{1 + 3} = \frac{15 + 5}{4} = \frac{20}{4} = 5
\]

Thus, the coordinates of point \( P \) are \( (4, 5) \).

Therefore, the point that should be placed on the segment to split it into a ratio of 1:3 is \( \boxed{(4, 5)} \).