Question

the centroid of triangle is (2,3) and two of its vertices are(5,6) and(-1,4).the third vertex is

Answer 1

Answer :

Let us take the 3rd vertex be (x, y)

The other two vertices are (5, 6)

and (- 1, 4)

Then, its centroid be at

( (x + 5 - 1)/3, (y + 6 + 4)/3 )

i.e., ( (x + 4)/3, (y + 10)/3 )

By the given condition,

( (x + 4)/3, (y + 10)/3 ) Ξ (2, 3)

Then,

(x + 4)/3 = 2

or, x + 4 = 6

or, x = 2

and

(y + 10)/3 = 3

or, y + 10 = 9

or, y = - 1

Therefore, the 3rd vertex be (2, - 1)

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Answer 2

Let the third vertex be x3 and y3 then,

$centroid(x) = \frac{x1 + x2 + x3}{3} = \frac{4 + x3}{3} \\ \\ 2 = \frac{4 + x3}{3} \\ \\ 6 = 4 + x3 \\ \\ x3 = 2$

and

$centroid(y) = \frac{y1 + y2 + y3}{3} \\ \\ 3= \frac{10 + y3}{3} \\ \\ 9 = 10 + y3 \\ \\ y3 = - 1$

vertex=(2,-1)