Question

The coordinates of A, B, C are (6, 3), (–3, 5) and (4, – 2) respectively and P is any point (x, y). Show that the ratio of the areas of triangle PBC and ABC is .
| $\frac{x+y-2}{7}$| [4 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

We have,

$\therefore Areaof\Delta PBC=\frac{1}{2}|(5x+6+4y)-(-3y+20-2x)|$

$\Rightarrow Areaof\Delta PBC=\frac{1}{2}|5x+6+4y+3y-20+2x|$

$\Rightarrow Areaof\Delta PBC=\frac{1}{2}|7x+7y-14|$

$\Rightarrow Areaof\Delta PBC=\frac{7}{2}|x+y-2|$

$\Rightarrow Areaof\Delta ABC=\frac{7}{2}|6+3-2|$

[ Replacing x by 6 and y = 3 in Area of $\Delta PBC$]

$\Rightarrow Areaof\Delta ABC=\frac{49}{2}$

$\therefore \frac{Areaof\Delta PBC}{Areaof\Delta ABC}=\frac{\frac{7}{2}|x+y-2|}{\frac{49}{2}}$

$\Rightarrow \frac{Areaof\Delta PBC}{Areaof\Delta ABC}=\frac{|x+y-2|}{7}$