SLOPES OF PARALLEL AND PERPENDICULAR LINES WORKSHEET

1.  The line joining the points A (-2, 3) and B (a, 5) is parallel to the line joining the points C (0, 5) and D (-2, 1). Find the value of a.

2.  The line joining the points A(0, 5) and B (4, 2) is perpendicular to the line joining the points C (-1, -2) and D (5, b). Find the value of b.

3.  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.

1. Answer :

Since the line joining the points AB and CD are parallel, the slope of those two lines will be equal.

Slope of AB :

A(-2, 3) ==> (x₁, y₁),  B(a, 5)  ==>  (x₂, y₂)

Slope (m₁)  =  (y₂ - y₁)/(x₂ - x₁)

  =  (5 - 3)/(a - (-2))

  =  2/(a + 2)

Slope of CD :

C(0 , 5) ==> (x₁, y₁),  D(-2, 1) ==> (x₂, y₂)

Slope (m₂)  =  (y₂ - y₁)/(x₂ - x₁)

  =  (1 - 5)/(-2 - 0)

  =  -4/(-2)

  =  2

Slope of AB  =  Slope of CD

2/(a + 2)  =  2

Multiply each side by (a + 2). 

2  =  2(a + 2)

2  =  2a + 4

Subtract 4 from each side. 

2 - 4  =  2a 

2a  =  -2

Divide each side by 2. 

a  =  -1

2. Answer :

Since the line joining the points AB and CD are perpendicular, the product of slopes will be equal to -1.

Slope of AB :

A (0, 5) ==> (x₁, y₁),  B (4, 2) ==>  (x₂, y₂)

Slope (m₁)  =  (y₂ - y₁)/(x₂ - x₁)

  =  (2 - 5)/(4 - 0)

  =  -3/4

Slope of CD :

C (-1, -2) ==> (x₁, y₁),  D (5 , b) ==>  (x₂, y₂)

Slope (m₂)  =  (y₂ - y₁)/(x₂ - x₁)

  =  (b - (-2))/(5 - (-1))

  =  (b + 2)/(5 + 1)

  =  (b + 2)/6

(Slope of AB) ⋅ (Slope of CD)  =  -1

(-3/4) ⋅ (b + 2)/6  =  -1

(b + 2)/8  =  1

Multiply each side by 8.

b + 2  =  8

Subtract 2 from each side. 

b  =  8 - 2

b  =  6

3. Answer :

Mid point of the side AB  =  M

Mid point  =  (x₁ + x₂)/2 ,  (y₁ + y₂)/2

=  [1 + (-2)]/2 , (8 + 4)/2

  =  (-1/2, 6)

Mid point of the side AC  =  N

Mid point  =  (x₁ + x₂)/2 ,  (y₁ + y₂)/2

=  [1 + 8]/2 , (8 - 5)/2

  =  (9/2, 3/2)

Slope of BC :

=   (y₂ - y₁)/(x₂ - x₁)

  =  (-5 - 4)/(8 + 2)

  =  -9/10  ------(1)

Slope of MN :

=   (y₂ - y₁)/(x₂ - x₁)

  = ((3/2) - 6)/((9/2) + (1/2)) 

  = (-9/2)/(10/2)

  =  -9/10  ------(2)

From (1) and (2),

slope of BC  =  slope of MN

So, the sides MN and BC are parallel.

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