SLOPE OF THE LINE PASSING THROUGH THE MIDPOINT OF THE LINE SEGMENT

Example 1 :

Find the slope of the line which passes through the origin and the midpoint of the line segment joining the points (0 ,- 4) and (8 , 0).

Solution :

Midpoint of the line segment  =  (x₁ + x₂)/2, (y₁ + y₂)/2

= (0 + 8)/2, (-4 + 0)/2

= (8/2, -4/2)

  = (4, -2)

Slope of the passing through the point (4, -2) and (0, 0) is

m = (y₂ - y₁)/(x₂ - x₁)

m = (0 + 2)/(0 - 4)

m = -2/4 = -1/2

Example 2 :

The side AB of a square ABCD is parallel to x-axis . Find the

(i) slope of AB (ii) slope of BC (iii) slope of the diagonal AC

Solution :

(i)  Slope of AB

The side AB is parallel to x-axis, hence the slope of AB is 0.

(ii) slope of BC 

The side BC is perpendicular to x-axis, hence its slope is undefined.

(iii) slope of the diagonal AC

Angle of inclination of the diagonal AC is 45

m  =  tan 45

m  =  1

Example 3 :

The side BC of an equilateral Δ ABC is parallel to x-axis. Find the slope of AB and the slope of BC

Solution :

The side BC is parallel to x-axis, hence its slope is undefined.

Since it is right triangle the angle of inclination is 60

m = tan 60

m = √3

Example 4 :

A line in the xy-plane is defined by the equation y = ax + b, where a and b are constants. The line passes through the points (0, -79) and (5, 91). What is the value of a + b ?

Solution :

The line passes through the points (0, -79) and (5, 91)

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

= (91 - (-79)) / (5 - 0)

= (91 + 79)/5

= 170/5

= 34

In the linear function y = ax + b, the value of a is 34.

y = 34x + b

From the given point (0, -79), the y-intercept is -79

y = 34x - 79

a = 34 and b = -79

a + b = 34 - 79

= 45

So, the value of a + b is 45.

Example 5 :

The table shows three values of x and their corresponding values of y, where p is a constant. There is a linear relationship between x and y. What is the slope of the line that represents this relationship in the xy-plane?

a) -57/4   b)  57/4     c)  -(p - 57)/4     d)  (2p + 57)/4

Solution :

The points are (-6, -114 - p), (-2, -57 - p) and (2, -p).

Slope of the line passes through the first two points,

= [(-57 - p) - (-114 - p)]/(-2 + 6)

= [-57 - p + 114 + p]/4

= 57/4

So, the slope of the line passes through the points is 57/4. Option b is correct.

Example 6 :

The graph of the equation 5x - 8y = k, where k is a constant, is a line in the xy-plane. The line has an x-intercept at (m , 0) and a y-intercept at (0, n), where m and n are constants. What is the value of m/n ?

a)  -8/5     b)  -5/8    c)  5/8    d)  8/5

Solution :

5x - 8y = k

x-intercept at (m, o)

5m - 8(0) = k

5m = k

m = k/5

y-intercept at (0, n)

5(0) - 8n = k

-8n = k

n = -k/8

m/n = (k/5) / (-k/8)

= -8/5

So, option a is correct.

Example 7 :

In the xy-plane, line m has an x-intercept at (a, b + 4), where a and b are constants. If the line m passes through the points (b, -11) and (b + 6, 7), what is the value of a ?

Solution :

Slope of the line passes through the points (b, -11) and (b + 6, 7)

= (7 - (-11)) / (b + 6 - b)

= (7 + 11) / 6

= 18/6

= 3

Equation of the line :

y - y₁ = m(x - x₁)

Using the point (b, -11), we get

y - (-11) = 3(x - b)

y + 11 = 3(x - b)

x-intercept is at (a, b + 4). The y-coordinate of the point at x-intercept is 0.

b + 4 = 0

b = -4

y + 11 = 3(x + 4)

(b + 6, 7) ==> (2, 7)

Applying the point (a, b + 4) that is (a, 0) we get

0 + 11 = 3(a + 4)

11 = 3a + 12

3a = 11 - 12

3a = -1

a = -1/3

So, the required value of a is -1/3.

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