Question

ThE centroid of a triangle is(2, 3) and two of its vertices are(5, 6) and(-1, 4) t
hen yhe third vertex of a triangle is?

Answer 1

Solution :

Centroid Of A Triangle -

For a triangle ABC with the coordinates being A ( x¹ , y¹ ) , B ( x², y² ) and C ( x³, y³ )

Centroid -

$\sf{ G = \dfrac{ ( x_1 + x_2 + x_3 ) }{ 3 } , \dfrac{ ( y_1 + y_2 + y_3 ) }{ 3 } }$

Let us assume that the third vertex of the triangle is (x,y)

> [ 5 - 1 + x]/3 ; [6 + 4 + y]/3

> ( x + 4)/3 ; ( y + 10)/3

(x+4)/3 = 2

> x + 4 = 6

> x = 2

(y+10)/3 = 3

> y = 9-10 = -1

Answer : The third vertex of the triangle is (2,-1)

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Answer 2

Given :-

Centroid = (2,3)

Two vertices = (5,6) and (-1,4)

To Find :-

Third vertex

Solution :-

Let the third vertex be (x₃,y₃)

Centroid formula = (x₁ + x₂ + x₃/3, y₁ + y₂ + y₃/3)

2,3 = [5 + (-1) + x₃/3, 6 + 4 + y₃/3]

2,3 = [5 - 1 + x₃/3, 10 + y₃/3]

2,3 = [4 + x₃/3, 10 + y₃/3]

Solving separately

2 = 4 + x₃/3

2 × 3 = 4 + x₃

6 = 4 + x₃

6 - 4 = x₃

2 = x₃

3 = 10 + y₃/3

3 × 3 = 10 +  y₃

9 = 10 + y₃

9 - 10 = y₃

-1 = y₃