Question

The midpoints pf the sides BC, CA,and AB of triangle ABC are D(3,4) E(8,9) and F(6,7). Find the coordinates of the vertices of the triangle

Answer 1

Answer: The vertices are A(11,12), B(1,2) and C(5,6).

Step-by-step explanation: As shown in the figure, let A(a,b), B(c,d) and C(e,f) be the vertices of ΔABC. Given that D(3,4), E(8,9) and F(6,7) are the mid-points of the sides BC, CA and AB respectively.

Since D is the mid-point of BC, so

$\dfrac{c+e}{2}=3~~\textup{and}~~\dfrac{d+f}{2}=4\\\\\Rightarrow c+e=6,~~d+f=8.$

Similarly, since F is the mid-point of AB, so

$c+a=12,~~d+b=14.$

∴ $a-e=6.~~b-f=6.$

Also, since E is the mid-point of AC, so

$a+e=16,~~b+f=18.$

Solving the above linear equations, we get

$a=11,~b=12,~c=1,~d=2,~e=5~\textup{and}~f=6.$

Thus, the vertices of ΔABC are A(11,12), B(1,2) and C(5,6).

Answer 2

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