A centroid of a triangle is the point where the three medians of the triangle meet. A median of a triangle is a line segment from one vertex to the mid point on the opposite side of the triangle.
The centroid G of the triangle with vertices A(x1, y1), B(x2 , y2 ) and C(x3 , y3) is
Let us look at some examples to understand how to find the missing vertex of a triangle when centroid is given
Example 1 :
Find the missing vertex of triangle whose two of the vertices are (0, 0), (-8, 0) and centroid is (-5, -2).
Solution :
Centroid of a triangle = (x1 + x2 + x3)/3, (y1 + y2 + y3)/3
Let the required vertex be C (a, b)
A(0, 0) B(-8, 0) and C (a, b)
x1 = 0, x2 = -8, x3 = a
y1 = 0, y2 = 0, y3 = b
(0 - 8 + a)/3 , (0 + 0 + b)/3 = (-5, -2)
(-8 + a)/3 , b/3 = (-5, -2)
Equating the x and y coordinates, we get
(-8 + a)/3 = -5 -8 + a = -15 a = -15 + 8 a = -7 |
b/3 = -2 b = -6 |
So the required vertex be (-7, -6).
Example 2 :
The centroid of a triangle ABC is (1, 1). Two of the vertices are A (3, -4), B (-4, 7). Find the coordinates of the third vertex.
Solution :
Centroid of a triangle = (x1 + x2 + x3)/3, (y1 + y2 + y3)/3
Let the required vertex be C (a, b)
A (3, -4), B (-4, 7) and C (a, b)
x1 = 3, x2 = -4, x3 = a
y1 = -4, y2 = 7, y3 = b
(3 - 4 + a)/3 , (-4 + 7 + b)/3 = (1, 1)
(-1 + a)/3 , b/3 = (1, 1)
Equating the x and y coordinates, we get
(-1 + a)/3 = 1 -1 + a = 3 a = 3 + 1 a = 4 |
b/3 = 1 b = 3 |
Hence the required vertex is (4, 3).
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