Question

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

A(-3, 1) B(-8, 5)

Answer 1

answers are

-3 1/3, 8/3

-4 2/3, 2 1/3

-4 1/3, -10

-10 1/3, 5

Answer 2

To find the coordinates of point \( X \) on segment \( AB \) that divides the segment in the ratio \( AX : BX = 1 : 2 \), we can use the section formula. The section formula states that if a point \( P \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) can be calculated as follows:

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

In this case, \( A(-3, 1) \) and \( B(-8, 5) \), and the ratio \( AX : BX = 1 : 2 \), which means \( m = 1 \) and \( n = 2 \).

### Plugging in the values:

- \( x_1 = -3 \)
- \( y_1 = 1 \)
- \( x_2 = -8 \)
- \( y_2 = 5 \)
- \( m = 1 \)
- \( n = 2 \)

### Calculate the x-coordinate of point \( X \):

\[
x = \frac{1 \cdot (-8) + 2 \cdot (-3)}{1 + 2} = \frac{-8 - 6}{3} = \frac{-14}{3} = -\frac{14}{3}
\]

### Calculate the y-coordinate of point \( X \):

\[
y = \frac{1 \cdot 5 + 2 \cdot 1}{1 + 2} = \frac{5 + 2}{3} = \frac{7}{3}
\]

### Therefore, the coordinates of point \( X \) are:

\[
X\left(-\frac{14}{3}, \frac{7}{3}\right)
\]

Thus, point \( X \) that partitions segment \( AB \) in the ratio \( 1 : 2 \) is located at:

\[
\boxed{\left(-\frac{14}{3}, \frac{7}{3}\right)}
\]

Answer 3

Let's clarify the calculation to find the coordinates of point \( X \) on segment \( AB \) that divides it in the ratio \( AX : BX = 1:2 \).

Given:
- Point \( A(-3, 1) \)
- Point \( B(-8, 5) \)
- The ratio \( m : n = 1 : 2 \) (where \( m = 1 \) and \( n = 2 \))

### Using the Section Formula

The coordinates \( (x, y) \) of point \( X \) can be calculated as follows:

\[
x = \frac{mx_2 + nx_1}{m+n} = \frac{1 \cdot (-8) + 2 \cdot (-3)}{1 + 2}
\]
\[
y = \frac{my_2 + ny_1}{m+n} = \frac{1 \cdot 5 + 2 \cdot 1}{1 + 2}
\]

### Calculating the x-coordinate

\[
x = \frac{-8 + (-6)}{3} = \frac{-14}{3} = -\frac{14}{3} \approx -4.67
\]

### Calculating the y-coordinate

\[
y = \frac{5 + 2}{3} = \frac{7}{3} \approx 2.33
\]

So the coordinates of point \( X \) calculated before are:

\[
X\left(-\frac{14}{3}, \frac{7}{3}\right) \approx (-4.67, 2.33)
\]

### Comparing with Your Options

1. **-3 \(\frac{1}{3}\), \(\frac{8}{3}\)** ≈ (-3, 2.67)
2. **-4 \(\frac{2}{3}\), \(2 \frac{1}{3}\)** ≈ (-4.67, 2.33)
3. **-4 \(\frac{1}{3}\), -10** ≈ (-4.33, -10)
4. **-10 \(\frac{1}{3}\), 5** ≈ (-10.33, 5)

From the values calculated, point \( X \) corresponds to the second option:

\[
\boxed{-4 \frac{2}{3}, 2 \frac{1}{3}}
\]

This matches. Thank you for your clarification, and apologies for any confusion in my earlier calculation.