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(−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(−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)
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(−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(−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)
Answers
Answered by
GPT-4o mini
To find the coordinates of point \( X \) that partitions segment \( AB \) in the ratio \( AX:BX = 1:2 \), we can use the section formula.
Let the coordinates of \( A \) be \( (x_1, y_1) \) and the coordinates of \( B \) be \( (x_2, y_2) \). According to the section formula, the coordinates of point \( X \) that divides the segment \( AB \) internally in the ratio \( m:n \) (where \( m = 1 \) and \( n = 2 \) in this case) can be found using the formulas:
\[
X_x = \frac{mx_2 + nx_1}{m+n}
\]
\[
X_y = \frac{my_2 + ny_1}{m+n}
\]
### Step 1: Identify coordinates of points A and B
Assuming point \( A = (x_1, y_1) \) and point \( B = (x_2, y_2) \) we need the actual coordinates to apply the formula. However, since you haven't provided these coordinates, let's assume you have them as follows:
Assuming:
- \( A(-4, -10) \) (i.e., coordinates of A)
- \( B(-3, 8) \) (i.e., coordinates of B)
### Step 2: Compute coordinates of point X
Using the section formula to find \( X \):
\[
X_x = \frac{1*(-3) + 2*(-4)}{1 + 2} = \frac{-3 - 8}{3} = \frac{-11}{3}
\]
\[
X_y = \frac{1*8 + 2*(-10)}{1 + 2} = \frac{8 - 20}{3} = \frac{-12}{3} = -4
\]
Thus, the coordinates of point \( X \) are:
\[
X\left( -\frac{11}{3}, -4 \right)
\]
However, if you have specific coordinates for points \( A \) and \( B \), please provide them, and I can confirm the calculations. Based on the responses you have provided, the values of \( X \) that you are looking for should align with the computed coordinates of \( X \).
Among the mentioned options, none explicitly match the simple division, hence feel free to specify the coordinates of \( A \) and \( B \) for accurate computation.
Let the coordinates of \( A \) be \( (x_1, y_1) \) and the coordinates of \( B \) be \( (x_2, y_2) \). According to the section formula, the coordinates of point \( X \) that divides the segment \( AB \) internally in the ratio \( m:n \) (where \( m = 1 \) and \( n = 2 \) in this case) can be found using the formulas:
\[
X_x = \frac{mx_2 + nx_1}{m+n}
\]
\[
X_y = \frac{my_2 + ny_1}{m+n}
\]
### Step 1: Identify coordinates of points A and B
Assuming point \( A = (x_1, y_1) \) and point \( B = (x_2, y_2) \) we need the actual coordinates to apply the formula. However, since you haven't provided these coordinates, let's assume you have them as follows:
Assuming:
- \( A(-4, -10) \) (i.e., coordinates of A)
- \( B(-3, 8) \) (i.e., coordinates of B)
### Step 2: Compute coordinates of point X
Using the section formula to find \( X \):
\[
X_x = \frac{1*(-3) + 2*(-4)}{1 + 2} = \frac{-3 - 8}{3} = \frac{-11}{3}
\]
\[
X_y = \frac{1*8 + 2*(-10)}{1 + 2} = \frac{8 - 20}{3} = \frac{-12}{3} = -4
\]
Thus, the coordinates of point \( X \) are:
\[
X\left( -\frac{11}{3}, -4 \right)
\]
However, if you have specific coordinates for points \( A \) and \( B \), please provide them, and I can confirm the calculations. Based on the responses you have provided, the values of \( X \) that you are looking for should align with the computed coordinates of \( X \).
Among the mentioned options, none explicitly match the simple division, hence feel free to specify the coordinates of \( A \) and \( B \) for accurate computation.