Long Questions (Question 2)

Question 2:
(c) the coordinates of the point of intersection between lines PQ and ST produced.


Solution:
(a)
$\begin{array}{l}R=\left(\frac{3+7}{2},\frac{2+6}{2}\right)\\ R=\left(5,4\right)\end{array}$

(b)
$\begin{array}{l}QR:RT=1:3\\ \text{Lets coordinates of }T=(x,y)\\ \left(\frac{\left(1\right)\left(x\right)+\left(3\right)\left(4\right)}{1+3},\frac{\left(1\right)\left(y\right)+\left(3\right)\left(5\right)}{1+3}\right)=\left(5,\text{ 4}\right)\\ \frac{x+12}{4}=5\\ x+12=20\\ x=8\\ \\ \frac{y+15}{4}=4\\ y+15=16\\ y=1\\ \\ T=(8,1)\end{array}$

(c)
$\begin{array}{l}\text{Gradient of }PQ=\frac{5-2}{4-3}=3\\ \text{Equation of }PQ,\\ y-2=3\left(x-3\right)\\ y-2=3x-9\\ y=3x-7----(1)\\ \\ \text{Gradient of }ST=\frac{6-1}{7-8}=-5\\ \text{Equation of }ST,\\ y-1=-5\left(x-8\right)\\ y-1=-5x+40\\ y=-5x+41----(2)\end{array}$