MCQ
If $\Delta_1$ is the area of the triangle formed by the centroid and two vertices of a  triangle, $\Delta_2$ is the area of the triangle formed by the mid-points of the sides of the  same triangle, then $\Delta_1 : \Delta_2 =$
  • A
    $3 : 4$
  • B
    $4 : 1$
  • $4 : 3$
  • D
    $2 : 1$

Answer

Correct option: C.
$4 : 3$
c
$\operatorname{Let} \mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right), \mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)$ and $\mathrm{C}\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right)$ be the vertices

of a $\Delta \mathrm{ABC},$ and let $\mathrm{G}$ be its centroid. Then, $\Delta_{1}=$ Area of $\Delta \mathrm{GBC}$

$\Rightarrow \Delta_{1}=\frac{\Delta}{3},$ where $\Delta$ is the area of $\Delta \mathrm{ABC}$

$\Delta_{2}=$ Area of triangle formed by the mid-points of the sides

$\Rightarrow \Delta_{2}=\frac{1}{4} \Delta$

$\therefore \Delta_{1}: \Delta_{2}=4: 3$

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