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Use the image to answer the question. Find the coordinates of point X on segment AB¯¯¯¯¯¯¯¯ that partitions the segment into the ratio AX:BX = 1:2 . (1 point) Responses X(−413,−10) upper X left parenthesis negative Start Fraction 13 over 3 End Fraction comma negative 10 right parenthesis X(−423,213) upper X left parenthesis negative Start Fraction 19 over 3 End Fraction comma Start Fraction 11 over 3 End Fraction right parenthesis 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 Skip to navigation

7 answers

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.

Let the coordinates of points \(A\) and \(B\) be \(A(x_1, y_1)\) and \(B(x_2, y_2)\). The section formula states that the coordinates of point \(X\) that divides the segment in the ratio \(m:n\) are given by:

\[ X_x = \frac{mx_2 + nx_1}{m+n} \] \[ X_y = \frac{my_2 + ny_1}{m+n} \]

In this case, \(m = 1\), \(n = 2\), and the coordinates of \(A\) and \(B\) can be assumed as \(A(-413, -10)\) and \(B(-423, 213)\).

Now plug these values into the formulas:

  1. Calculate \(X_x\):

\[ X_x = \frac{1(-423) + 2(-413)}{1+2} = \frac{-423 - 826}{3} = \frac{-1249}{3} = -416.33 \]

  1. Calculate \(X_y\):

\[ X_y = \frac{1(213) + 2(-10)}{1+2} = \frac{213 - 20}{3} = \frac{193}{3} \approx 64.33 \]

Thus, the coordinates of point \(X\) that partitions segment \(AB\) in the ratio \(1:2\) are approximately \((-416.33, 64.33)\).

However, given the response options you provided, they appear to be specific rational fractions or decimal notations. Thus, checking the options, we can find which matches our calculated values.

Among the listed choices:

  • \( X(-\frac{13}{3}, -10) \approx (-4.33, -10)\)
  • \( X(-\frac{19}{3}, \frac{11}{3}) \approx (-6.33, 3.67)\)
  • \( X(-\frac{31}{3}, 5) \approx (-10.33, 5)\)
  • \( X(-\frac{10}{3}, \frac{8}{3}) \approx (-3.33, 2.67)\)

None of these match the coordinates we calculated. It seems there may have been an error in the coordinates of points A and B that were initially provided. Double-checking those might lead to a more accurate conclusion. If the coordinates of A and B are correct, you would need to ensure the individual ratios and calculations were correctly undertaken.

Feel free to provide the exact coordinates of A and B for further refinement on the outcome.

B's coordinates are (-8, 5) and A's are (-3, 1).

Given the coordinates of points A and B are \( A(-3, 1) \) and \( B(-8, 5) \), we want to find the coordinates of point \( X \) that partitions segment \( AB \) in the ratio \( AX:BX = 1:2 \).

We'll use the section formula again with \( m = 1 \) and \( n = 2 \):

The formulas for the coordinates of point \( X \) are:

\[ X_x = \frac{mx_2 + nx_1}{m+n} \] \[ X_y = \frac{my_2 + ny_1}{m+n} \]

Substituting the values:

  • \( m = 1 \)
  • \( n = 2 \)
  • Coordinates of \( A(-3, 1) \) \( (x_1 = -3, y_1 = 1) \)
  • Coordinates of \( B(-8, 5) \) \( (x_2 = -8, y_2 = 5) \)

Let's calculate \( X_x \) first:

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

Next, calculate \( X_y \):

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

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

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

Now, we'll compare this result with the response options you provided earlier:

  1. \( X(-\frac{13}{3}, -10) \)
  2. \( X(-\frac{19}{3}, \frac{11}{3}) \)
  3. \( X(-\frac{31}{3}, 5) \)
  4. \( X(-\frac{10}{3}, \frac{8}{3}) \)

None of the options appears to match the calculated coordinates of \( X\left(-\frac{14}{3}, \frac{7}{3}\right) \).

Let me summarize:

  • The coordinates of point \( X \), which partitions segment \( AB \) in the ratio \( 1:2 \), are \( X\left(-\frac{14}{3}, \frac{7}{3}\right) \).
  • Please verify the available options or if there’s additional information.

If there are any discrepancies in the values provided, please let me know!

can you convert the answer to a mixed number?

Sure! The coordinates of point \(X\) are \(-\frac{14}{3}\) for the x-coordinate and \(\frac{7}{3}\) for the y-coordinate.

  1. For \(-\frac{14}{3}\):

    • Divide \(14\) by \(3\): \[ 14 \div 3 = 4 \quad \text{(whole part)} \]
    • The remainder is \(2\) because \(3 \times 4 = 12\), and \(14 - 12 = 2\).
    • So, \(-\frac{14}{3}\) can be expressed as: \[ -4 \frac{2}{3} \]

  2. For \(\frac{7}{3}\):

    • Divide \(7\) by \(3\): \[ 7 \div 3 = 2 \quad \text{(whole part)} \]
    • The remainder is \(1\) because \(3 \times 2 = 6\), and \(7 - 6 = 1\).
    • So, \(\frac{7}{3}\) can be expressed as: \[ 2 \frac{1}{3} \]

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

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

THANK YOU!! 😄😄

You're welcome! 😊 I'm glad I could help. If you have any more questions or need further assistance, feel free to ask. Have a great day! 🌟