Question

$A(6,3),B(-3,5),C(4,-2)$ and $D(x,3x)$ are four points. If $\Delta DBC:\Delta ABC=1:2,$ then x is equal to

• A. $\frac{11}{8}$
• B. $\frac{8}{11}$
• C. 3
• D. none of these

Answer 1

The correct option is A

$\frac{11}{8}$

The area of a triangle with vertices $D(x,3x),B(-3,5)$ and $C(4,-2)$ is given below:

Area of $\Delta DBC=\frac{1}{2}\left\{x\right(5+2)-3(-2-3x)+4(3x-5\left)\right\}$

$\Rightarrow$ Area of $\Delta DBC=(14x-7)$ sq. units.

Similarly, the area of a triangle with vertices $A(6,3),B(-3,5)$ and $C(4,-2)$ is given below :

$\Delta ABC=\frac{1}{2}\left\{6\right(5+2)-3(-2-3)+4(3-5\left)\right\}$

$\Rightarrow \Delta DBC=\frac{49}{2}$ sq units

Given : $\Delta DBC:\Delta ABC=1:2$

$\frac{2(14x-7)}{49}=\frac{1}{2}$

$\Rightarrow 8x-4=7$

$\Rightarrow x=\frac{11}{8}$