Question

The point (2,3), (x,y) and (3,-2) are vertices of a triangle. If the centroid of this triangle is again (x,y), find (x,y)?

Answer 1

To find centroid:

If (x₁, y₁), (x₂, y₂), (x₃, y₃) are the vertices of any triangle, then its centroid is at

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

Now we solve the given problem:

The vertices of the given triangle are (2, 3), (x, y) and (3, - 2).

Using the above formula, we can find that the centroid of the triangle is at

( (2 + x + 3)/3, (3 + y - 2)/3 )

i.e., ( (x + 5)/3, (y + 1)/3 )

By the given condition,

( (x + 5)/3, (y + 1)/3 ) Ξ (x, y)

Then (x + 5)/3 = x and (y + 1)/3 = y

or, x + 5 = 3x or, y + 1 = 3y

or, 2x = 5 or, 2y = 1

or, x = 5/2 or, y = 1/2

So (x, y) Ξ (5/2, 1/2).

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Answer 2

Given :

Three vertices of triangle as poitns = (2,3), (x,y) and (3,-2)

Coordinate of centroid = (x ,y)

To find :

Value of x and y

Solution :

• We know that,

Centroid is given by formula

(x, y) = [ (x1 + x2 + x3/3), (y1 + y2 + y3/3) ]

• Therefore,

(x, y) = [ (2 + x + 3 / 3), (3 + y - 2 / 3) ]

Comparing both sides

x = 2 + x + 3 / 3

3x = 5 + x

2x = 5

x = 5/2

• Also,

y = 3 + y - 2 / 3

3y = 1 + y

2y = 1

y = 1/2

• Hence,

( x, y ) = ( 5/2, 1/2 )