Question

find the coordinates of the point $\frac{7}{10}$ of the way from a to b. the coordinates of the point $\frac{7}{10}$ of the way from a to b are (type an ordered pair.)

Explanation:

1. Assume the coordinates of point \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\):

• From the graph (not clearly - shown in the text, but for a general formula), if we want to find the coordinates of a point \(P\) that is \(t=\frac{7}{10}\) of the way from \(A\) to \(B\), we use the section - formula.
• The \(x\) - coordinate of the point \(P\) is given by \(x=x_1 + t(x_2 - x_1)\).
• The \(y\) - coordinate of the point \(P\) is given by \(y=y_1 + t(y_2 - y_1)\).
• Let's assume \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). Then \(x=x_1+\frac{7}{10}(x_2 - x_1)=\frac{10x_1+7x_2 - 7x_1}{10}=\frac{3x_1 + 7x_2}{10}\).
• And \(y=y_1+\frac{7}{10}(y_2 - y_1)=\frac{10y_1+7y_2 - 7y_1}{10}=\frac{3y_1 + 7y_2}{10}\).

1. If we assume \(A=(0,0)\) and \(B=(10,10)\) (for illustration purposes, since the actual coordinates of \(A\) and \(B\) are not given in text):

• For the \(x\) - coordinate: \(x = 0+\frac{7}{10}(10 - 0)=7\).
• For the \(y\) - coordinate: \(y = 0+\frac{7}{10}(10 - 0)=7\).
• In general, if \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\), the coordinates of the point \(\frac{7}{10}\) of the way from \(A\) to \(B\) are \((x_1+\frac{7}{10}(x_2 - x_1),y_1+\frac{7}{10}(y_2 - y_1))\).

Since the actual coordinates of \(A\) and \(B\) are not provided in the text, if we assume \(A=(0,0)\) and \(B=(10,10)\) (a simple case for demonstration), the answer is \((7,7)\). But the general formula for the coordinates of the point \(P\) that is \(\frac{7}{10}\) of the way from \(A=(x_1,y_1)\) to \(B=(x_2,y_2)\) is \((x_1+\frac{7}{10}(x_2 - x_1),y_1+\frac{7}{10}(y_2 - y_1))\).

Step1: Recall the section - formula

The formula for a point \(P\) that divides the line - segment joining \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) in the ratio \(t:1 - t\) is \((x_1 + t(x_2 - x_1),y_1 + t(y_2 - y_1))\). Here \(t=\frac{7}{10}\).

Step2: Calculate the \(x\) - coordinate

\(x=x_1+\frac{7}{10}(x_2 - x_1)\).

Step3: Calculate the \(y\) - coordinate

\(y=y_1+\frac{7}{10}(y_2 - y_1)\).

Answer:

\((x_1+\frac{7}{10}(x_2 - x_1),y_1+\frac{7}{10}(y_2 - y_1))\) (general form). If \(A=(0,0)\) and \(B=(10,10)\), the answer is \((7,7)\)