Question

A (6, 3), B (−3, 5), C (4, −2) and D (x, 3x) are four points. If ∆ DBC : ∆ ABC = 1 : 2, then x is equal to
(a) 11/8
(b) 8/11
(c) 3
(d) none of these

Answer 1

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

$\Rightarrow$Area of ∆DBC = $\left(14x-7\right)\mathrm{sq}\mathrm{units}$

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

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

$\Rightarrow$∆ABC = $\frac{49}{2}$sq units

Given:
∆DBC:∆ABC = 1:2

$$\begin{gathered} \frac{2\left(14x-7\right)}{49}=\frac{1}{2} \\ \Rightarrow 8x-4=7 \\ \Rightarrow x=\frac{11}{8} \end{gathered}$$