Question

2 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:

Step1: Recall section - formula

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

Step2: Assume coordinates of \(Q\) and \(S\)

Let \(Q(x_1,y_1)\) be the starting - point and \(S(x_2,y_2)\) be the ending - point of the line - segment \(\overline{QS}\). Suppose \(Q=(a,b)\) and \(S=(c,d)\) (from the graph, we need to read the coordinates of \(Q\) and \(S\)). Let's assume \(Q=( - 4,-4)\) and \(S=(6,6)\) (by observing the grid).

Step3: Calculate \(x\) - coordinate of \(R\)

Using the formula \(x=\frac{mx_2+nx_1}{m + n}\), substitute \(m = 3\), \(n = 5\), \(x_1=-4\), and \(x_2 = 6\).
\[x=\frac{3\times6+5\times(-4)}{3 + 5}=\frac{18-20}{8}=\frac{-2}{8}=-\frac{1}{4}\]

Step4: Calculate \(y\) - coordinate of \(R\)

Using the formula \(y=\frac{my_2+ny_1}{m + n}\), substitute \(m = 3\), \(n = 5\), \(y_1=-4\), and \(y_2 = 6\).
\[y=\frac{3\times6+5\times(-4)}{3 + 5}=\frac{18 - 20}{8}=-\frac{1}{4}\]

Answer:

The coordinates of \(R\) are \((-\frac{1}{4},-\frac{1}{4})\)