Question

Find the coordinates of point X on segment AB that partitions the segment into the ratio AX: BX= 1:2

Answer 1

To find the coordinates of point \( X \) that partitions the segment \( AB \) in the ratio \( AX: BX = 1:2 \), we can use the section formula from coordinate geometry.

Let \( A(x_1, y_1) \) and \( B(x_2, y_2) \) be the coordinates of points \( A \) and \( B \) respectively.

If the segment \( AB \) is divided in the ratio \( m:n = 1:2 \), the coordinates of point \( X \) can be found using the formula:

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

For our case, \( m = 1 \) and \( n = 2 \). Plugging these values into the section formula gives us:

\[
X\left( \frac{1 \cdot x_2 + 2 \cdot x_1}{1+2}, \frac{1 \cdot y_2 + 2 \cdot y_1}{1+2} \right)
\]

Simplifying this:

\[
X\left( \frac{x_2 + 2x_1}{3}, \frac{y_2 + 2y_1}{3} \right)
\]

Thus, the coordinates of point \( X \) that partitions the segment \( AB \) in the ratio \( 1:2 \) are given by:

\[
X\left( \frac{x_2 + 2x_1}{3}, \frac{y_2 + 2y_1}{3} \right)
\]

You can substitute the specific coordinates of points \( A \) and \( B \) to find the exact location of point \( X \).