Question

13. A triangle has vertices at (6 , 7), (2 , -9) and (-4 , 1). Find the slopes of its
medians.

Answer 1

Let a triangle has vertices A ≡ (6, 7) , B ≡ (2, -9) and C ≡ (-4, 1) and D , E and F are the midpoint of BC , CA , AB respectively. So, AD , BE and CF are the medians of triangle ABC.
So, first of all we have to find points D , E and F
D is the midpoint of BC .
so, D ≡ [ (2 -4)/2 , (-9 + 1)/2 ] [from midpoint section formula]
D ≡ (-1 , -4)

E is the midpoint of CA.
so, E ≡ [ (-4 + 6)/2 , (1 + 7)/2 ]
E ≡ (1 , 4)

F is the midpoint of AB
F ≡ [ (6 + 2)/2 , (7 - 9)/2 ]
F ≡ (4 , -1)

Now, A ≡ ( 6, 7) and D≡ (-1 , -4)
Slope of median AD = (-4 - 7)/(-1 - 6) = 11/7

similarly, slope of median BE = (4 + 9)/(1 - 2) = -13
slope of median CF = (1 + 1)/(4 + 4) = 1/4

Answer 2

Solution :

Let A( 6,7 ), B(2,-9) and C(-4,1)

are three vertices of a triangle .

D , E , F are midpoints of BC ,

CA and AB respectively .

i ) D is the midpoint of

B(2,-9)= ( x1 , y1 ) and

C(-4,1) = ( x2 , y2 )

Coordinates of D

= [ ( 2 - 4 )/2 , ( -9 + 1 )/2 ]

= ( -2/2 , -8/2 )

= ( -1 , -4 )

Similarly ,

E = ( 1 , 4 ) , and F = ( 4 , -1 )

ii) Slope of a line joining

A(6,7) = ( x1 , y1 ) and

D( -1 , -4 ) = ( x2 , y2 ) median

= ( y2 - y1 )/( x2 - x1 )

= ( -4 -7 )/( -1 -6 )

m = ( -11 )/( -7 )

m = 11/7

iii ) slope of the median

C( -4 , 1 ) and F( 4, -1 )

m = ( -1 - 1 )/( 4 + 4 )

= (-2)/8

m = -1/4

iv ) slope of median

B( 2 , -9 ) and E(1, 4 )

m = ( 4+9 )/( 1-2 )

= ( 13 )/( -1)

m = -13

••••