Question

The midpoints of the sides BC, CA and AB of a ΔABC are D(3, 4), E(8, 9) and F(6, 7) respectively. Find the coordinates of the vertices of the triangle. [CBSE 2017]

Answer 1

Let the coordinates of A, B, C be $(x_1,y_1),(x_2,y_2)\mathrm{and}(x_3,y_3)\mathrm{respectively}.$
Because D is the mid-point of BC, using mid-point formula, we have
$$\begin{gathered} \frac{x_2+x_3}{2}=3\mathrm{and}\frac{y_2+y_3}{2}=4 \\ \Rightarrow x_2+x_3=6\mathrm{and}{y}_2+y_3=8.....\left(\mathrm{i}\right) \end{gathered}$$
Similarly, E is the mid point of AC. Using mid-point formula, we have;
$$\begin{gathered} \frac{x_1+x_3}{2}=8\mathrm{and}\frac{y_1+y_3}{2}=9 \\ \Rightarrow x_1+x_3=16\mathrm{and}{y}_1+y_3=18.....\left(\mathrm{ii}\right) \end{gathered}$$
Again, F is the mid point of AB. Using mid point formula, we have
$$\begin{gathered} \frac{x_1+x_2}{2}=6\mathrm{and}\frac{y_1+y_2}{2}=7 \\ \Rightarrow x_1+x_2=12\mathrm{and}{y}_1+y_2=14.....\left(\mathrm{iii}\right) \end{gathered}$$
Adding (i), (ii) and (iii), we get
$$\begin{gathered} 2\left(x_1+x_2+x_3\right)=34\mathrm{and}2\left(y_1+y_2+y_3\right)=40 \\ \Rightarrow x_1+x_2+x_3=17\mathrm{and}{\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3=20.....\left(\mathrm{iv}\right) \end{gathered}$$
On solving equation (iv), using equations (i), (ii) and (iii), we get
$$\begin{gathered} x_1=11 \\ {x}_2=1 \\ {x}_3=5 \end{gathered}$$
Similarly,
$$\begin{gathered} y_1=12 \\ {y}_2=2 \\ {y}_3=6 \end{gathered}$$
Hence, the points are: A(11,12), B(1,2) and C(5,6).