Question

Abc is a triangle whose vertices are (3,4) (-2,-1) and (5,3). If g is the centroid and bdcg is a parallelogram then find the coordinates of d

Answer 1

Answer:

( 9, 6)

Step-by-step explanation:

Hi,

Given vertices of triangle ABC

A(3, 4) , B(-2, -1) and C(5, 3)

Centroid of triangle whose vertices are(Xa, Ya), (Xb, Yb) and (Xc, Yc)

is given by ( Xa + Xb + Xc/3, Ya + Yb + Yc/3)

So, centroid of triangle ABC will be G(3-2+5/3, 4-1+3/3)

= G ( 2, 2)

Given that BCDG is a parallelogram

Let the coordinates of D be (h, k)

But, we know that diagonals in a parallelogram bisect each other

So, midpoint of BD = Midpoint of CG

Midpoint of BD is (h - 2/2 , k - 1/2)

Midpoint of CG is ( 5 + 2/2, 3 + 2/2)

h - 2 = 5 + 2

h = 9

k - 1 = 3 + 2

k = 6

Hence, the coordinates of point D are (9, 6)

Hope, it helps !

Answer 2

A (3, 4) , B(-2, -1) and C (5, 3)

then, centroid, G = [ (3 - 2 + 5)/3, (4 - 1 + 3)/3] = (2, 2)

[ note : formula of centroid is $\left[\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right]$ ]

a/c to question, BDCG is a parallelogram. but we know, diagonals of parallelogram bisect each other.

so, midpoint of diagonal BD = midpoint of diagonal CG

midpoint of diagonal BD = [(-2 + x)/2, (-1 + y)/2 ]


midpoint of diagonal CG = [(5 + 2)/2, (3 + 2)/2] = (7/2, 5/2)

so, (7/2, 5/2) = [ (-2 + x)/2, (-1 + y)/2 ]

7/2 = (-2 + x)/2 => x = 9
5/2 = (-1+ y)/2 => y = 6

hence, D = (9, 6)