Question

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0).

Answer 1

The given vertices of triangle ABC are A=( 0,0,6 ) , B=( 0,4,0 ) and C=( 6,0,0 ) .

Let AD, BE, CF be the medians of triangle ABC.

Since AD is a median, D is the midpoint of BC. Therefore, D divides BC in the ratio 1:1 .

Using section formula,

D( x,y,z )=( m x 2 +n x 1 m+n , m y 2 +n y 1 m+n , m z 2 +n z 1 m+n ) (1)

x= ( 1⋅6+1⋅0 ) 1+1 = 6 2 =3

y= ( 1⋅0+1⋅4 ) 1+1 = 4 2 =2

z= ( 1⋅0+1⋅0 ) 1+1 = 0 2 =0

Therefore, the coordinates of D are ( 3,2,0 ) .

Distance d between two points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) is,

d= ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 (2)

Using the distance formula to find AD,

AD= ( 3−0 ) 2 + ( 2−0 ) 2 + ( 0−6 ) 2 = 3 2 + 2 2 + ( −6 ) 2 = 9+4+36 = 49 =7 units

Point E divides AC in ratio 1:1. Thus, using section formula,

a= ( 1⋅6+1⋅0 ) 1+1 = 6 2 =3

b= ( 1⋅0+1⋅0 ) 1+1 = 0+0 2 =0

c= ( 1⋅0+1⋅6 ) 1+1 = 6 2 =3

Thus, the coordinates of E are ( 3,0,3 ) .

Using the distance formula to find length of BE,

BE= ( 3−0 ) 2 + ( 0−4 ) 2 + ( 3−0 ) 2 = 3 2 + ( −4 ) 2 + 3 2 = 9+16+9 = 34  units

F divides AB in the ratio 1:1. Therefore, using section formula,

p= ( 1⋅0+1⋅0 ) 1+1 = 0 2 =0

q= ( 1⋅4+1⋅0 ) 1+1 = 4+0 2 = 4 2 =2

r= ( 1⋅0+1⋅6 ) 1+1 = 6 2 =3

Therefore, the coordinates of F are ( 0,2,3 ) .

Using the distance formula to find length of CF,

CF= ( 0−6 ) 2 + ( 2−0 ) 2 + ( 3−0 ) 2 = ( −6 ) 2 + 2 2 + 3 2 = 36+4+9 = 49   =7 units

Thus, the lengths of the medians AD, BE, CF of triangle ABC are 7 units , 34  units and 7 units respectively.