HOW TO CHECK IF TWO LINES ARE PARALLEL WITH COORDINATES

Question :

The vertices of triangle ABC are A(1, 8), B(-2, 4), C(8, -5). If M and N are the midpoints of AB and AC respectively, find the slope of MN and hence verify that MN is parallel to BC.

Answer :

If two lines are parallel, then their slopes are equal.

In the given triangle ABC, if MN is parallel to BC, then slopes of MN and BC must be equal

Midpoint of the side AB = M.

Midpoint of the side AC = N.

If a line joining the two points (x₁, y₁) and (x₂, y₂), then the formula to find midpoint :

 A(1, 8), B(-2, 4), C(8, -5)

Find the slope of MN :

Slope of MN = (y₂ - y₁)/(x₂ - x₁)

Here (x₁, y₁) = M(-1/2, 6) and (x₂, y₂) = N(9/2, 3/2).

y₂ - y₁ = 3/2 - 6

= -9/2

x₂ - x₁ = 9/2 + 1/2

= 5

Slope of MN = (-9/2)/5

Slope of MN = -9/10 ----(1)

Find the slope of BC :

Slope of BC = (y₂ - y₁)/(x₂ - x₁)

Here (x₁, y₁) = B(-2, 4) and (x₂, y₂) = C(8, -5).

Slope of BC = (-5 - 4)/(8 + 2)

Slope of BC = -9/10 ----(2)

From (1) and (2),

Slope of MN = Slope of BC

Hence the sides BC And MN are parallel.

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