If two vertices of a triangle are $(3, 7)$ and $(- 1, 5)$ and its centroid is $(1, 3)$ . Find the coordinates of the third vertex.

(1, - 3)

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(1, 3)

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(- 1, 3)

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None of these

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Solution

The correct option is B $(1, - 3)$
Centroid of a triangle with vertices ${(x}_{1}, y_{1}); {(x}_{2}, y_{2})$ and ${(x}_{3}, y_{3})$ is calculated by the formula
$(\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})$
Let the third vertex of the triangle be $(x, y)$
So, centroid $= (\frac{3 - 1 + x}{3}, \frac{7 + 5 + y}{3}) = (1, 3)$
$\Rightarrow (\frac{2 + x}{3}, \frac{12 + y}{3}) = (1, 3)$
$\Rightarrow 2 + x = 3; 12 + y = 9$
$\Rightarrow x = 1; y = - 3$
Hence the third vertex of the triangle is $(1, - 3)$

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If two vertices of a triangle are (3,7) and (−1,5) and its centroid is (1,3). Find the coordinates of the third vertex.

If two vertices of a triangle are $(3, 7)$ and $(- 1, 5)$ and its centroid is $(1, 3)$ . Find the coordinates of the third vertex.