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(x_{1}, y_{1}), B(x_{2} , y_{2} ) and C(x_{3} , y_{3}) 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 = (x_{1 }+ x_{2 }+ x_{3})/3, (y_{1 }+ y_{2 }+ y_{3})/3

Let the required vertex be C (a, b)

A(0, 0) B(-8, 0) and C (a, b)

x_{1 } = 0, x_{2} = -8, x_{3} = a

y_{1 } = 0, y_{2} = 0, y_{3} = 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 = (x_{1 }+ x_{2 }+ x_{3})/3, (y_{1 }+ y_{2 }+ y_{3})/3

Let the required vertex be C (a, b)

A (3, -4), B (-4, 7) and C (a, b)

x_{1 } = 3, x_{2} = -4, x_{3} = a

y_{1 } = -4, y_{2} = 7, y_{3} = 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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