Question

given directed line segment $overline{qs}$, find the coordinates of $r$ such that the ratio of $qr$ to $rs$ is $3:5$. plot point $r$.

Explanation:

1. Let the coordinates of \(Q=(x_1,y_1)\) and \(S=(x_2,y_2)\). The formula to find the coordinates of a point \(R=(x,y)\) that divides the line - segment \(\overline{QS}\) in the ratio \(m:n\) is given by:

• \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 3\) and \(n = 5\).
• First, assume we know the coordinates of \(Q=(x_1,y_1)\) and \(S=(x_2,y_2)\) from the graph. Let \(Q=(x_1,y_1)\) and \(S=(x_2,y_2)\).
• For the \(x\) - coordinate of \(R\):
• \(x=\frac{3x_2 + 5x_1}{3+5}=\frac{3x_2+5x_1}{8}\).
• For the \(y\) - coordinate of \(R\):
• \(y=\frac{3y_2 + 5y_1}{3+5}=\frac{3y_2+5y_1}{8}\).

1. Suppose \(Q=(x_1,y_1)=(a,b)\) and \(S=(x_2,y_2)=(c,d)\) after reading from the graph:

• Calculate \(x=\frac{3c + 5a}{8}\).
• Calculate \(y=\frac{3d + 5b}{8}\).

Since the graph is not clear - enough to read the exact coordinates of \(Q\) and \(S\), if we assume \(Q=(x_1,y_1)\) and \(S=(x_2,y_2)\) are known, the coordinates of \(R\) are \((\frac{3x_2+5x_1}{8},\frac{3y_2+5y_1}{8})\).

Step1: Recall the section - formula

The formula for a point \(R\) that divides the line - segment joining \(Q(x_1,y_1)\) and \(S(x_2,y_2)\) in the ratio \(m:n\) is \(R=(\frac{mx_2+nx_1}{m + n},\frac{my_2+ny_1}{m + n})\). Here \(m = 3\) and \(n = 5\).

Step2: Substitute \(m = 3\) and \(n = 5\) into the formula

We get \(x=\frac{3x_2+5x_1}{8}\) and \(y=\frac{3y_2+5y_1}{8}\).

Answer:

The coordinates of \(R\) are \((\frac{3x_2+5x_1}{8},\frac{3y_2+5y_1}{8})\), where \((x_1,y_1)\) are the coordinates of \(Q\) and \((x_2,y_2)\) are the coordinates of \(S\).