Question

Given A(−1,2)
and B(3,6)
, find the coordinates of point X
on AB¯¯¯¯¯¯¯¯
such that the ratio of AX
to XB
is 3:5.(1 point)
Responses

(−14,312)
Start Fraction 3 over 2 End Fraction

(125,425)
Start Fraction 1 over 3 End Fraction

(12,312)
Start Fraction 2 over 3 End Fraction

(112,112)

Answer 1

To find the coordinates of point \( X \) on line segment \( AB \) such that the ratio of \( AX \) to \( XB \) is \( 3:5 \), we can use the section formula.

Let \( A(-1, 2) \) and \( B(3, 6) \).

According to the section formula, if a point \( X \) divides the line segment \( AB \) in the ratio \( m:n \), then the coordinates of point \( X \) can be found using the following formulas:

\[
X_x = \frac{nA_x + mB_x}{m+n}
\]
\[
X_y = \frac{nA_y + mB_y}{m+n}
\]

Here, \( m = 3 \), \( n = 5 \) (since we want \( AX : XB = 3 : 5 \)).

Substituting the coordinates of points \( A \) and \( B \):

\[
X_x = \frac{5(-1) + 3(3)}{3 + 5} = \frac{-5 + 9}{8} = \frac{4}{8} = \frac{1}{2}
\]

\[
X_y = \frac{5(2) + 3(6)}{3 + 5} = \frac{10 + 18}{8} = \frac{28}{8} = \frac{7}{2}
\]

Thus, the coordinates of point \( X \) are \( \left( \frac{1}{2}, \frac{7}{2} \right) \).

Since none of the provided responses exactly match, but in converted improper fractions, \( \left( \frac{1}{2}, \frac{7}{2} \right) \) can be written as \( (0.5, 3.5) \). None of the decimal values provided match.

After double-checking, if you would like the response in ratio form, you can represent:

- \( X = (1/2, 7/2) \)

This does not match any specific options you've provided.

If you can confirm the answers or provide revisions needed, I'll be ready to assist further.