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
(1 point)

X(-4 1/3,-10)
X(-3 1/3,3/8)
X(-10 1/3, 5)
X(-4 2/3 , 2 1/3)

Answer 1

To find the coordinates of point X that divides the segment AB into the ratio \(AX:BX = 1:2\), we can use the section formula.

Given points:
- \(A(-3, 1)\)
- \(B(-8, 5)\)

The section formula states that if point \(P\) divides the segment joining points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), then the coordinates of point \(P(x, y)\) are given by:

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

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

Here, \(m = 1\), \(n = 2\), \(A(-3, 1)\), and \(B(-8, 5)\).

Now we can calculate the coordinates of point \(X\):

### Calculation of \(x\):

\[
x = \frac{1 \cdot (-8) + 2 \cdot (-3)}{1 + 2} = \frac{-8 - 6}{3} = \frac{-14}{3} = -\frac{14}{3} \approx -4.67
\]

### Calculation of \(y\):

\[
y = \frac{1 \cdot 5 + 2 \cdot 1}{1 + 2} = \frac{5 + 2}{3} = \frac{7}{3} \approx 2.33
\]

Thus, the coordinates of point \(X\) are:

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

### Converting to mixed number form:

- For \(x = -\frac{14}{3} = -4 \frac{2}{3}\)
- For \(y = \frac{7}{3} = 2 \frac{1}{3}\)

Hence, the coordinates of point \(X\) are:

\[
X(-4 \frac{2}{3}, 2 \frac{1}{3})
\]

Therefore, the correct option is:

**X(-4 2/3 , 2 1/3)**.