Question
Find point X along segment AB that makes AX¯¯¯¯¯¯¯¯:BX¯¯¯¯¯¯¯¯ 1:2. (1 point) Responses X(2,4) cap x times open paren 2 comma 4 close paren X(1,2) cap x times open paren 1 comma 2 close paren X(1.5,3) cap x times open paren 1 point 5 comma 3 close paren X(0.75,1.5). B 3,7 A 0,0
Answers
Answered by
GPT-4o mini
To find point \(X\) along segment \(AB\) such that the ratio \(AX:BX = 1:2\), we can use the section formula. Let's denote the coordinates of point \(A\) as \(A(0, 0)\) and point \(B\) as \(B(3, 7)\).
From the ratio \(AX:BX = 1:2\), we can denote the ratio as \(m:n = 1:2\), where \(m = 1\) and \(n = 2\).
Using the section formula, the coordinates of point \(X\) that divides the segment \(AB\) in the ratio \(m:n\) are given by:
\[
X = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Here, \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (3, 7)\).
Plugging in the values:
\[
X_x = \frac{1 \cdot 3 + 2 \cdot 0}{1 + 2} = \frac{3}{3} = 1
\]
\[
X_y = \frac{1 \cdot 7 + 2 \cdot 0}{1 + 2} = \frac{7}{3} \approx 2.33
\]
So, coordinates of point \(X\) are:
\[
X(1, \frac{7}{3}) \approx X(1, 2.33)
\]
However, \(2.33\) is not one of your answer options. Let's check the expression we wrote for \(X_y\):
\[
X_y = \frac{my_2 + ny_1}{m+n} \Rightarrow = \frac{1 \cdot 7 + 2 \cdot 0}{3} = \frac{7}{3} = 2.33 = 2.3333\ldots
\]
Checking the provided options, it appears that the coordinates \(X(1.5, 3)\) best fits the criteria, however, it should be \(X(1, \frac{7}{3})\) perfect coordinates as they physically represent the ratio \(AX:BX = 1:2\).
Thus, the coordinates of point \(X\) that divides segment \(AB\) in the required ratio are:
\[
X(1, \frac{7}{3})
\]
If rounded down from the fourth point above, an approximate point, it is most apt to make \(X(1,2)\) your best functional cap close to a straight line since the example is somewhat vague near listed answer choices.
From the ratio \(AX:BX = 1:2\), we can denote the ratio as \(m:n = 1:2\), where \(m = 1\) and \(n = 2\).
Using the section formula, the coordinates of point \(X\) that divides the segment \(AB\) in the ratio \(m:n\) are given by:
\[
X = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Here, \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (3, 7)\).
Plugging in the values:
\[
X_x = \frac{1 \cdot 3 + 2 \cdot 0}{1 + 2} = \frac{3}{3} = 1
\]
\[
X_y = \frac{1 \cdot 7 + 2 \cdot 0}{1 + 2} = \frac{7}{3} \approx 2.33
\]
So, coordinates of point \(X\) are:
\[
X(1, \frac{7}{3}) \approx X(1, 2.33)
\]
However, \(2.33\) is not one of your answer options. Let's check the expression we wrote for \(X_y\):
\[
X_y = \frac{my_2 + ny_1}{m+n} \Rightarrow = \frac{1 \cdot 7 + 2 \cdot 0}{3} = \frac{7}{3} = 2.33 = 2.3333\ldots
\]
Checking the provided options, it appears that the coordinates \(X(1.5, 3)\) best fits the criteria, however, it should be \(X(1, \frac{7}{3})\) perfect coordinates as they physically represent the ratio \(AX:BX = 1:2\).
Thus, the coordinates of point \(X\) that divides segment \(AB\) in the required ratio are:
\[
X(1, \frac{7}{3})
\]
If rounded down from the fourth point above, an approximate point, it is most apt to make \(X(1,2)\) your best functional cap close to a straight line since the example is somewhat vague near listed answer choices.