Question

find the coordinates of the point that partitions ab¯ in the ratio 3:7. the coordinates of the point that partitions ab¯ in the ratio 3:7 are (type an ordered pair).

Explanation:

1. First, assume the coordinates of point \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). Let's say from the graph (although not clearly - visible in the provided image, we use the section - formula). The section formula for a point \(P=(x,y)\) that divides the line - segment joining \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) in the ratio \(m:n\) is given by:

• \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\), where \(m = 3\) and \(n = 7\).
• Let's assume \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) are the endpoints of the line - segment \(\overline{AB}\).
• For the \(x\) - coordinate of the point \(P\) that divides \(\overline{AB}\) in the ratio \(3:7\):
• \(x=\frac{3x_2 + 7x_1}{3 + 7}=\frac{3x_2+7x_1}{10}\).
• For the \(y\) - coordinate of the point \(P\) that divides \(\overline{AB}\) in the ratio \(3:7\):
• \(y=\frac{3y_2 + 7y_1}{3 + 7}=\frac{3y_2+7y_1}{10}\).
• However, since the actual coordinates of \(A\) and \(B\) are not given in the text part (only the graph is provided but not clear enough to read exact coordinates), if we assume \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) are known, we can calculate as follows.
• Let \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\).
• Suppose \(x_1 = a\), \(x_2 = b\), \(y_1 = c\), and \(y_2 = d\).
• The \(x\) - coordinate of the dividing point: \(x=\frac{3b + 7a}{10}\).
• The \(y\) - coordinate of the dividing point: \(y=\frac{3d+7c}{10}\).

1. Let's assume for the sake of example, if \(A=(1,2)\) and \(B=(9,8)\):

• For the \(x\) - coordinate:
• \(x=\frac{3\times9 + 7\times1}{10}=\frac{27 + 7}{10}=\frac{34}{10}=3.4\).
• For the \(y\) - coordinate:
• \(y=\frac{3\times8+7\times2}{10}=\frac{24 + 14}{10}=\frac{38}{10}=3.8\).

Since we don't have the actual coordinates of \(A\) and \(B\) from the graph clearly, the general formula for the coordinates of the point that divides the line - segment \(\overline{AB}\) in the ratio \(3:7\) is \((\frac{3x_2+7x_1}{10},\frac{3y_2+7y_1}{10})\).

Step1: Recall section - formula

The formula for a point dividing a line - segment in ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\), \(y=\frac{my_2+ny_1}{m + n}\) with \(m = 3\) and \(n = 7\).

Step2: Express \(x\) and \(y\) coordinates

We get \(x=\frac{3x_2+7x_1}{10}\) and \(y=\frac{3y_2+7y_1}{10}\).

Answer:

\((\frac{3x_2+7x_1}{10},\frac{3y_2+7y_1}{10})\)