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Question
ABC is a triangle and G(4, 3) is the centroid of the triangle. If A = (1, 3), B = (4, b) and C = (a, 1), find ‘a’ and ‘b’. Find the length of side BC.
Solution
The coordinates of the vertices of ΔABC are A(1, 3), B(4, b) and C(a, 1).
It is known that A(x1, y1), B(x2, y2) and C(x3, y3) are vertices of a triangle, then
The coordinates of centroid G = `((x_1 + x_2 + x_3)/3, (y_1 + y_2 + y_3)/3)`
Thus, the coordinates of the centroid of ABC are
`((1+4+a).3, (3+b+1)/3) = ((5+a)/3, (4+b)/3)`
It is given that the coordinates of the centroid are G(4, 3).
Therefore, we have
`(5+a)/3 = 4`
`5 + a = 12`
a = 7
`(4+ b)/3 = 3`
4 + b = 9
b = 5
Thus, the coordinates of B and C are (4, 5) and (7, 1) respectively.
Using distance formula, we have
`BC = sqrt((7-4)^2 + (1 - 5)^2)`
`= sqrt(9 + 16)`
`= sqrt25`
= 5 units
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