The mid- points of the sides of a triangle, ABC are: (-2,3,5), (4,-1,7) and (6,5,3) Find the co- ordinates - Maths - Introduction to Three Dimensional Geometry

Question

The mid- points of the sides of a triangle, ABC are: (-2,3,5),
(4,-1,7) and (6,5,3) Find the co- ordinates of A,B,C

Ajanta Trivedi · Accepted Answer

let the coordinates of A, B and C are
(x1 ,y1,z1);(x2 ,y2,z2) and (x3 ,y3,z3) respectively
let the mid-point of AB, BC and CA be D, E and F respectively
let the coordinates of mid-points are D (-2,3,5),
E(4,-1,7);F(6,5,3)
therefore
x1+x22=-2 ;y1+y22=3 and z1+z22=5⇒x1+x2=-4 (1); y1+y2=6 (2) and
z1+z2=10 (3)similarlyx2+x3=4*2=8 (4); y2+y3=-1*2=-2 (5) and
z2+z3=7*2=14 (6)x3+x1=6*2=12 (7); y3+y1=5*2=10 (8) and z3+z1=3*2=6
(9)
subtracting eq(1) from eq(4): x3-x1=12 ; x3+x1=12 [from
eq(7)2x3=24⇒x3=12and x1=12-x3=12-12⇒x1=0from eq(1)
x2=-4-x1=-4-0=-4therefore x1=0;x2=-4 and x3=12subtracting eq(2)
from eq(5) :y3-y1=-8y3+y1=102y3=2⇒y3=1y1=10-y3=10-1=9from
eq(2): y2=6-y1=6-9=-3therefore y1=9;y2=-3 and y3=1
subtracting eq(3) from eq(6)z3-z1=4z3+z1=62
z3=10⇒z3=5z1=6-z3=6-5=1from eq(7): z2=10-z1=10-1=9therefore
z1=1;z2=9 and z3=5
thus the coordinates of A are (0,9,1) and B(-4,-3,9) and
C(12,1,5)

hope this helps you

The mid- points of the sides of a triangle, ABC are: (-2,3,5), (4,-1,7) and (6,5,3). Find the co- ordinates of A,B,C.