Question

1. given directed line - segment jl below, find the coordinates of k such that the ratio of jk to jl is 7:8. plot point k.

given a directed line segment ab with coordinates a(x1,y1) and b(x2,y2), if point p partitions ab such that the ratio of ap to ab is k, then the coordinates of p are: ((k(x2 - x1))+x1,k(y2 - y1))+y1)
*important: the ratio must be written as a part to whole in fraction form!

1. given ab with a(3, - 1) and b(8,14), if p lies on ab such that the ratio of ap to pb is 2:3, find the coordinates of p.
2. given mp with m(-12, - 5) and p(9, - 12), if n lies on mp such that the ratio of mn to np is 3:4, find the coordinates of n.
3. given xz with x(1,9) and z(5, - 11), if y lies on xz such that the ratio of xy to xz is 1:4, find the coordinates of y.

© gina wilson (all things algebra®, llc). 2014 - 201

Explanation:

Step1: Recall the section - formula

The formula for finding the coordinates of 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 \(y=\frac{my_2+ny_1}{m + n}\), which is equivalent to \(x=k(x_2 - x_1)+x_1\) and \(y=k(y_2 - y_1)+y_1\) where \(k=\frac{m}{m + n}\).

Step2: Solve problem 9

Given \(A(3,-1)\) and \(B(8,14)\) and the ratio \(AP:PB = 2:3\), then \(k=\frac{2}{2 + 3}=\frac{2}{5}\).
For the \(x\) - coordinate of \(P\):
\(x=\frac{2\times8+3\times3}{2 + 3}=\frac{16 + 9}{5}=\frac{25}{5}=5\)
For the \(y\) - coordinate of \(P\):
\(y=\frac{2\times14+3\times(-1)}{2 + 3}=\frac{28-3}{5}=\frac{25}{5}=5\)
So the coordinates of \(P\) are \((5,5)\).

Step3: Solve problem 10

Given \(M(-12,-5)\) and \(P(9,-12)\) and the ratio \(MN:NP = 3:4\), then \(k=\frac{3}{3 + 4}=\frac{3}{7}\).
For the \(x\) - coordinate of \(N\):
\(x=\frac{3\times9+4\times(-12)}{3 + 4}=\frac{27-48}{7}=\frac{-21}{7}=-3\)
For the \(y\) - coordinate of \(N\):
\(y=\frac{3\times(-12)+4\times(-5)}{3 + 4}=\frac{-36-20}{7}=\frac{-56}{7}=-8\)
So the coordinates of \(N\) are \((-3,-8)\).

Step4: Solve problem 11

Given \(X(1,9)\) and \(Z(5,-11)\) and the ratio \(XY:XZ = 1:4\), then \(k=\frac{1}{4}\).
For the \(x\) - coordinate of \(Y\):
\(x=\frac{1\times5+3\times1}{4}=\frac{5 + 3}{4}=2\)
For the \(y\) - coordinate of \(Y\):
\(y=\frac{1\times(-11)+3\times9}{4}=\frac{-11 + 27}{4}=\frac{16}{4}=4\)
So the coordinates of \(Y\) are \((2,4)\).

Answer:

1. \((5,5)\)
2. \((-3,-8)\)
3. \((2,4)\)