the vertices of a triangle ABC are A( - 3 , 2) B (- 1 ,- 4 ) and C (5 , 2) to if m and n are the midpoints of AB and AC respectively show that 2 MN is equals to BC
Answers
The vertices of a triangle ABC are A( - 3 , 2) B (- 1 ,- 4 ) and C (5 , 2) to it m and n are the midpoints of AB and AC respectively.
Given,
A( - 3 , 2) B (- 1 ,- 4 ) and C (5 , 2)
M is the mid-point of AB
N is the mid-point of AC
Mid-point formula is given by,
(x, y) = { (x1 + x2) / 2 , (y1 + y2) / 2 }
The mid-point of AB is
M (x, y) = { ( -3 -1 ) / 2 , ( 2 - 4 ) / 2 } = ( -2, -1)
M( -2, -1)
The mid-point of AC is
N (x, y) = { ( -3 + 5 ) / 2 , ( 2 + 2 ) / 2 } = ( 1, 2)
N( 1, 2)
Distance formula is given by,
d = √ [ (x2 - x1)^2 + (y2 - y1)^2 ]
The distance between B and C
BC = √ [ (5 +1)^2 + (5 +4)^2 ]
= √ [ 6^2 + 9^2 ]
∴ BC = √ (117)
The distance between M and N
MN = √ [ (1 +2)^2 + (2 +1)^2 ]
= √ [ 3^2 + 3^2 ]
∴ MN = √ (18)
∴ 2 MN = BC
Hence it is shown that 2 MN = BC