Question

Find the coordinates of point X on segment AB that partitions the segment into the ratio AX: BX= 1:2 B(-8,5) A(-3,2)=×

Answer 1

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

The section formula states that if a point \( X \) divides the segment \( AB \) in the ratio \( m:n \), the coordinates of \( X \) can be calculated using the following formula:

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

Here, \( A \) has coordinates \( (-3, 2) \) and \( B \) has coordinates \( (-8, 5) \). In this case, \( m = 1 \) and \( n = 2 \).

### Step 1: Identify coordinates of points A and B
- \( A = (x_1, y_1) = (-3, 2) \)
- \( B = (x_2, y_2) = (-8, 5) \)

### Step 2: Substitute values into the section formula
\[
X\left(\frac{1 \cdot (-8) + 2 \cdot (-3)}{1 + 2}, \frac{1 \cdot 5 + 2 \cdot 2}{1 + 2}\right)
\]

### Step 3: Calculate the x-coordinate of X
\[
X_x = \frac{-8 + (-6)}{3} = \frac{-14}{3} = -\frac{14}{3}
\]

### Step 4: Calculate the y-coordinate of X
\[
X_y = \frac{5 + 4}{3} = \frac{9}{3} = 3
\]

### Step 5: Combine the coordinates
Thus, the coordinates of point \( X \) are:
\[
X = \left(-\frac{14}{3}, 3\right)
\]

So the final answer is:
\[
X \left(-\frac{14}{3}, 3\right)
\]