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in Coordinate Geometry by (28.9k points)
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A (3, 2) and B (-2, 1) are two vertices of a triangle ABC whose centroid G has the coordinates (\(\frac{5}3\), - \(\frac{1}3\)). Find the coordinates of the third vertex C of the triangle.

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Let the vertex of the triangle be A(3, 2), B(-2, 1) and C(x, y) 

Let the centroid be G(\(\frac{5}3\) , \(\frac{-1}3\)), as it is given that centroid is given at origin.

We know that centroid of a triangle for (x1 , y1), (x2 , y2) and (x3 , y3) is 

G(x, y) = \( (\frac{x_1+x_2+x_3}3,\frac{y_1+y_2+y_3}3)\)

For given coordinates A(3, 2), B(-2, 1) and C(x, y)

 G(\(\frac{5}3\) , \(\frac{-1}3\)) = \( (\frac{3-2+x}3,\frac{2+1+y}3)\)

Solving for x and y, 

-3 + 2 +x =5 and 2 + 1 + y = -1 

∴ x = 6 and y = -4 

Hence, 

the coordinate of third vertex is C(6, -4).

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