# 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.

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

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

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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