Question

name moriah hollis sec. 1 introduction to geometry - points, lanes and planes homework 4 topics 6&7 partitioning a line segment 1. given g(1,2) and k(8,12). find the coordinates of p that partitions gk in the ratio of 3:2. 2. given the points m(-3,-4) and t(5,0), find the coordinates of the point q on directed line segment mt that partitions mt in the ratio 2:3. 3. point a has coordinates (7,2). point b has coordinates (2,-10). find the coordinates of point p that partition ab in the ratio 5:1. 4. given the points a(-1,2) and b(7,8), find the coordinates of the point p on the directed line segment ab that partitions ab in the ratio 1:3. plot p along with segment ab.

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 1

Given \(G(1,2)\) and \(K(8,12)\) and the ratio \(m:n = 3:2\).
\(x=\frac{3\times8+2\times1}{3 + 2}=\frac{24 + 2}{5}=\frac{26}{5}=5.2\)
\(y=\frac{3\times12+2\times2}{3 + 2}=\frac{36+4}{5}=\frac{40}{5}=8\)
The coordinates of \(P\) are \((5.2,8)\).

Step3: Solve problem 2

Given \(M(-3,-4)\) and \(T(5,0)\) and the ratio \(m:n = 2:3\).
\(x=\frac{2\times5+3\times(-3)}{2 + 3}=\frac{10-9}{5}=\frac{1}{5}=0.2\)
\(y=\frac{2\times0+3\times(-4)}{2 + 3}=\frac{0 - 12}{5}=-2.4\)
The coordinates of \(Q\) are \((0.2,-2.4)\).

Step4: Solve problem 3

Given \(A(7,2)\) and \(B(2,-10)\) and the ratio \(m:n = 5:1\).
\(x=\frac{5\times2+1\times7}{5 + 1}=\frac{10 + 7}{6}=\frac{17}{6}\approx2.83\)
\(y=\frac{5\times(-10)+1\times2}{5 + 1}=\frac{-50 + 2}{6}=\frac{-48}{6}=-8\)
The coordinates of \(P\) are \((\frac{17}{6},-8)\).

Step5: Solve problem 4

Given \(A(-1,2)\) and \(B(7,8)\) and the ratio \(m:n = 1:3\).
\(x=\frac{1\times7+3\times(-1)}{1 + 3}=\frac{7-3}{4}=1\)
\(y=\frac{1\times8+3\times2}{1 + 3}=\frac{8 + 6}{4}=\frac{14}{4}=3.5\)
The coordinates of \(P\) are \((1,3.5)\).

Answer:

1. \(P(5.2,8)\)
2. \(Q(0.2,-2.4)\)
3. \(P(\frac{17}{6},-8)\)
4. \(P(1,3.5)\)