Question

$ABC$ is a triangle with vertices $A(-1,4),B(6,-2)\&c(-2,4)$,$D,\hspace{0.17em}E\&F$are the points which divide each $AB,BC\&CA$, internally in the ratio $3:1$. Then, the centroid of $△DEF$ is

• A. $(3,6)$
• B. $(1,2)$
• C. $(4,8)$
• D. $(-3,6)$
• E. None of the above

Answer 1

The correct option is B

$(1,2)$

Explanation for the correct option:

Step 1. Finding coordinate of $D,\hspace{0.17em}E\&F$.

Let $(x_1,y_1),(x_2,y_2),\&(x_3,y_3)$ be the coordinates of $D,\hspace{0.17em}E\&F$.

Given that $D,\hspace{0.17em}E\&F$ divide $AB,BC\&CA$ internally in the ratio of $3:1$

Using formulae to find the coordinate of $D,\hspace{0.17em}E\&F$

$\begin{array}{rcl}{x}_1& =& \frac{\left[3\right(6)-1(1\left)\right]}{4}\\ & =& \frac{17}{4}\\ y_1& =& \frac{[-2(3)+4(1\left)\right]}{4}\\ & =& \frac{-1}{2}\end{array}$$$\begin{gathered} {\mathit{x}}_{\mathbf{1}}\mathbf{=}\frac{\mathbf{m}{\mathbf{X}}_{\mathbf{1}}\mathbf{+}\mathbf{n}{\mathbf{X}}_{\mathbf{2}}}{\mathbf{2}} \\ \mathbf{}{\mathit{y}}_{\mathbf{1}}\mathbf{=}\frac{\mathbf{m}{\mathbf{Y}}_{\mathbf{1}}\mathbf{+}\mathbf{n}{\mathbf{Y}}_{\mathbf{2}}}{\mathbf{2}} \end{gathered}$$

Similarly,

$\begin{array}{rcl}{x}_2& =& 0,y_2=\frac{5}{2}\\ x_3& =& \frac{-5}{4},y_3=4\end{array}$

Step 2. Finding point of centroid.

Assume $(x,y)$ be the centroid of the triangle.

Then,

$$\begin{gathered} x=\left(\frac{x_1+x_2+x_3}{3}\right) \\ y=\left(\frac{y_1+y_2+y_3}{3}\right) \end{gathered}$$

Now put all the value

Hence,

$\begin{array}{rcl}x& =& \frac{1}{3}\left[\right(\frac{17}{4})+0+(\frac{-5}{4}\left)\right]\\ & =& 1\\ y& =& \frac{1}{3}\left[\right(\frac{-1}{2})+(\frac{5}{2})+4]\\ & =& 2\end{array}$

Therefore, the coordinate of the centroid is $(1,2).$
Hence, the correct option is $\left(B\right)$