Question

The mid - points of the sides of a triangle are $D(6,1),E(3,5)$ and $F(-1,-2)$, then the vertex opposite to $D$ is

• A. $(-4,2)$
• B. $(-4,5)$
• C. $(2,5)$
• D. $(10,8)$

Answer 1

The correct option is A

$(-4,2)$

Explanation for the correct option:

Find the required point.

We have given that, the mid points of the sides of a triangle are$D(6,1),E(3,5)$ and $F(-1,-2)$, then $D,E,F,$and vertex opposite to$D$ forms a parallelogram.

We know that in a parallelogram, diagonals bisect each other, they have same midpoint.

Let$A(x,y)$ be the vertex opposite to$D$

So,

$\left(\frac{6+x}{2},\frac{1+y}{2}\right)=\left(\frac{3-1}{2},\frac{5-2}{2}\right)$

$\left(\frac{6+x}{2},\frac{1+y}{2}\right)=\left(1,\frac{3}{2}\right)$

Therefore, $\frac{6+x}{2}=1$

$$\begin{gathered} \Rightarrow 6+x=2 \\ \Rightarrow x=2-6 \\ \Rightarrow x=-4 \end{gathered}$$

and also,$\frac{1+y}{2}=\frac{3}{2}$

$$\begin{gathered} \Rightarrow 1+y=3 \\ \Rightarrow y=2 \end{gathered}$$

So the vertex opposite$D$ to is $(-4,2)$

Hence, option (A) is the correct answer.