Question

Use the image to answer the question.

Segment upper A upper B is graphed on a coordinate plane. The x-axis ranges from negative 9 to 1 in increments of 1. The y-axis ranges from 0 to 6 in increments of 1.
A= (-3, 1)
B= (-8, 5)
Find the coordinates of point X on segment AB¯¯¯¯¯¯¯¯ that partitions the segment into the ratio AX:BX = 1:2 .

(1 point)
Responses

X(−1013,5)
upper X left parenthesis negative Start Fraction 31 over 3 End Fraction comma 5 right parenthesis

X(−313,83)
upper X left parenthesis negative Start Fraction 10 over 3 End Fraction comma Start Fraction 8 over 3 End Fraction right parenthesis

X(−413,−10)
upper X left parenthesis negative Start Fraction 13 over 3 End Fraction comma negative 10 right parenthesis

X(−423,213)

Answer 1

To find the coordinates of point X that partitions segment AB in the ratio AX:BX = 1:2, we can use the section formula. According to the section formula, if point X divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, the coordinates of point X are given by:

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

In this case, we have:

- A = (-3, 1)
- B = (-8, 5)
- The ratio m:n = 1:2 (so m = 1, n = 2)

Using this information, we can calculate the coordinates of point X:

1. Calculate the x-coordinate of X:

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

2. Calculate the y-coordinate of X:

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

Thus, the coordinates of point X are:

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

However, let's evaluate the corresponding fractions given in the options to see if one of them matches.

- \(-\frac{14}{3}\) is equal to \(-4.\overline{6}\) or approximately -4.67.
- \(\frac{7}{3}\) is equal to approximately 2.33.

Now from the options listed:

1. \(X\left(-\frac{31}{3}, 5\right)\)
2. \(X\left(-\frac{10}{3}, \frac{8}{3}\right)\)
3. \(X\left(-\frac{13}{3}, -10\right)\)
4. \(X\left(-\frac{4}{3}, 2.13\right)\)

None of the options given in your list correspond to the ratio-derived coordinates of \(X\). Please check the possible answers or the calculations to find the correct option. Based on the calculation, the answer should logically be:

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

This value seems absent from the options provided.