Everything Maths Grade 11

16.5 CHAPTER 16. GEOMETRY

1

2

3

1 2 3

θ

Δy

Δx

x

y

1

2

3

1 2 3 4

f(x) = 4x− 4

g(x) = 2x− 2

θg θf

x

y

(a) (b)

Figure 16.4: (a) A line makes an angle θ with the x-axis. (b) The angle is dependent on the gradient. If
the gradient of f is mfand the gradient of g is mgthen mf> mgand θf> θg.

Firstly, we note that if the gradient changes, thenthe value of θ changes (Figure 16.4(b)), so we suspect
that the inclination of aline is related to the gradient. We know that thegradient is a ratio of a change
in the y-direction to a change inthe x-direction.

m =

Δy

Δx

But, in Figure 16.4(a) wesee that

tan θ =

Δy

Δx

∴ m = tan θ

For example, to find theinclination of the line y = x, we know m = 1

∴ tan θ = 1

∴ θ = 45◦

Exercise 16 - 3

1. Find the equations of the following lines

(a) through points (−1;3) and (1;4)

(b) through points (7;−3) and (0;4)

(c) parallel to y =^12 x + 3 passing through (−1;3)

(d) perpendicular to y =−^12 x + 3 passing through (−1;2)

(e) perpendicular to 2 y + x = 6 passing through the origin

2. Find the inclination of the following lines

(a) y = 2x− 3

(b) y =^13 x− 7

(c) 4 y = 3x + 8

(d) y =−^23 x + 3 (Hint: if m is negative θ must be in the second quadrant)

(e) 3 y + x− 3 = 0

3. Show that the line y = k for any constant k is parallel to the x-axis. (Hint: Show that the
   inclination of this line is 0 ◦.)