Question

1. 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).
2. given directed line - segment (overline{km}), find the coordinates of (l) such that the ratio of (kl) to (km) is (1:3). plot point (l).
3. given directed line - segment (overline{cd}), if point (e) divides (cd) three - fourths of the way from (c) to (d), find the coordinates of (e), then plot (e).

Explanation:

Step1: Recall the section - formula

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

Step2: Solve problem 2

Let the coordinates of \(Q=(x_1,y_1)\) and \(S=(x_2,y_2)\). The ratio of \(QR\) to \(RS\) is \(3:5\), so \(m = 3\) and \(n = 5\). The coordinates of \(R\) are \(x_R=\frac{3x_2+5x_1}{3 + 5}\) and \(y_R=\frac{3y_2+5y_1}{3 + 5}\). First, identify the coordinates of \(Q\) and \(S\) from the graph. Then substitute into the formula to get the coordinates of \(R\).

Step3: Solve problem 3

Let the coordinates of \(K=(x_1,y_1)\) and \(M=(x_2,y_2)\). The ratio of \(KL\) to \(KM\) is \(1:3\), so \(m = 1\) and \(n = 2\) (since if \(KL:KM=1:3\), then \(KL:LM = 1:2\)). Using the section - formula, \(x_L=\frac{1x_2+2x_1}{1 + 2}\) and \(y_L=\frac{1y_2+2y_1}{1 + 2}\). Identify the coordinates of \(K\) and \(M\) from the graph and substitute.

Step4: Solve problem 4

Let the coordinates of \(C=(x_1,y_1)\) and \(D=(x_2,y_2)\). Point \(E\) divides \(CD\) three - fourths of the way from \(C\) to \(D\), so \(m = 3\) and \(n = 1\). Using the section - formula, \(x_E=\frac{3x_2+1x_1}{3 + 1}\) and \(y_E=\frac{3y_2+1y_1}{3 + 1}\). Identify the coordinates of \(C\) and \(D\) from the graph and substitute.

Answer:

For problem 2: After identifying \(Q\) and \(S\) coordinates and substituting into \(x_R=\frac{3x_2+5x_1}{8}\), \(y_R=\frac{3y_2+5y_1}{8}\), get the coordinates of \(R\).
For problem 3: After identifying \(K\) and \(M\) coordinates and substituting into \(x_L=\frac{x_2 + 2x_1}{3}\), \(y_L=\frac{y_2+2y_1}{3}\), get the coordinates of \(L\).
For problem 4: After identifying \(C\) and \(D\) coordinates and substituting into \(x_E=\frac{3x_2+x_1}{4}\), \(y_E=\frac{3y_2+y_1}{4}\), get the coordinates of \(E\).