Question

1. find point z on $overline{rs}$ with r (-1, 1) and s (7, 2) such that the ratio of rz to zs is 1:3.
2. find the point g on $overline{ab}$ with a (-3, 5) and b (5, 0) such that the ratio of ag to gb is 3:2.
3. find the point e on $overline{uv}$ with u (2, -4) and v (4, -3) such that the ratio of ue to ev is 3:4.

Explanation:

1.

Step1: Recall the section - formula

The formula for finding 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 \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(R(-1,1)\), \(S(7,2)\), and \(m = 1\), \(n = 3\).

Step2: Calculate the x - coordinate of \(Z\)

\[x=\frac{1\times7+3\times(-1)}{1 + 3}=\frac{7-3}{4}=\frac{4}{4}=1\]

Step3: Calculate the y - coordinate of \(Z\)

\[y=\frac{1\times2+3\times1}{1 + 3}=\frac{2 + 3}{4}=\frac{5}{4}=1.25\]

Step1: Apply the section - formula

For points \(A(-3,5)\) and \(B(5,0)\) with \(m = 3\) and \(n = 2\), use \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\).

Step2: Calculate the x - coordinate of \(G\)

\[x=\frac{3\times5+2\times(-3)}{3 + 2}=\frac{15-6}{5}=\frac{9}{5}=1.8\]

Step3: Calculate the y - coordinate of \(G\)

\[y=\frac{3\times0+2\times5}{3 + 2}=\frac{0 + 10}{5}=2\]

Step1: Use the section - formula

Given \(U(2,-4)\), \(V(4,-3)\), \(m = 3\), and \(n = 4\), use \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\).

Step2: Calculate the x - coordinate of \(E\)

\[x=\frac{3\times4+4\times2}{3 + 4}=\frac{12 + 8}{7}=\frac{20}{7}\approx2.86\]

Step3: Calculate the y - coordinate of \(E\)

\[y=\frac{3\times(-3)+4\times(-4)}{3 + 4}=\frac{-9-16}{7}=\frac{-25}{7}\approx - 3.57\]

Answer:

\(Z(1,1.25)\)

2.