Question

A segment 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 split the segment from \( (2, 5) \) to \( (10, 5) \) in a ratio of \( 1:3 \), we can determine the point that divides the segment accordingly.

The coordinates of the endpoints of the segment are:
- Point A: \( (2, 5) \)
- Point B: \( (10, 5) \)

The formula to find a point \( P \) that divides the segment from \( A \) to \( B \) in the ratio \( m:n \) is given by:

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

In this case, \( m = 1 \) and \( n = 3 \).

Substituting the coordinates of points A and B into the formula:

- \( x_1 = 2 \)
- \( y_1 = 5 \)
- \( x_2 = 10 \)
- \( y_2 = 5 \)

Now plugging in the values,

\[
P(x, y) = \left( \frac{1 \cdot 10 + 3 \cdot 2}{1+3}, \frac{1 \cdot 5 + 3 \cdot 5}{1+3} \right)
\]

Calculating the x-coordinate:

\[
x = \frac{10 + 6}{4} = \frac{16}{4} = 4
\]

Calculating the y-coordinate:

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

Thus, the point \( P \) that divides the segment in a ratio of \( 1:3 \) is \( (4, 5) \).