# Lines

Geometry is the most practical branch of mathematics that deals with shapes and sizes of figures and their properties. A **line** is one-dimensional. That is, a line has length, but no width or height. A line extends forever in both directions. A line is uniquely determined by two points. These two points can be placed anywhere in a 2D plane. A line is also categorised into different types.

## What is a Line?

- A line can be defined as a straight set of points that extend in opposite directions
- has no ends in both directions(infinite)
- has no thickness
- it is one-dimensional

## What is a Line segment?

- A line segment is part of a line
- It has a beginning point and an ending point

## What is a Ray?

- A ray is a part of a line that has one endpoint (i.e. starting point)
- Extends in one direction endlessly.

## Number Line

The number line is well-suited to support informal thinking strategies of students because of its inherent linearity. In contrast to blocks or counters with a “set-representation” orientation, young children naturally recognize marks on a number line as visual representations of the mental images that most people have when they learn to count and develop an understanding of number relationships. It is important to note the difference between an “open number line” (shown below) and a ruler with its predetermined markings and scale.

### An open number line

The open line allows students to partition, or subdivide, the space as they see fit, and as they may need, given the problem context at hand. In other words, the number line above could be a starting point for any variety of number representations, two of which are shown below: the distance from zero to 1, or the distance from zero to 100. Once a second point on a number line has been identified, the number line moves from being an open number line, to a closed number line. In addition, the open number line allows for flexibility in extending counting strategies from counting by ones, for example, to counting by tens or hundreds all on the same sized open number lines.

### Closed number lines

Ray is part of a line; has one endpoint and continues on and on in one direction As it is one end is non-terminating, length of the line cannot be measured.

The definition of a ray depends upon the notion of between points on a line. It follows that rays exist only for geometries for which this notion exists

We can name a ray using its starting point and one other point that is

on the ray: this is ray AC or ray. Or, we can name a ray using a lowercase letter: this is ray r.

## Types of Line

In Geometry there are basically four types of lines. They are:

**Horizontal Lines:**When a line moves from left to right direction, it is horizontal.**Vertical Lines:**When a runs from top to bottom it is vertical.**Parallel Lines:**When two lines don’t meet each other at any point, even at infinity, then they are parallel.**Perpendicular Lines:**When two lines meet each other at an angle of 90 degrees, they are perpendicular to each other.

Other types of Lines are:

#### 1.Tangent lines

The Tangent is a straight line which just touches the curve at a given point. The normal is a straight line which is perpendicular to the tangent. To calculate the equations of these lines we shall make use of the fact that the equation of a straight line passing through the point with coordinates (x1, y1) and having gradient m is given by

\(\frac{y-y_{1}}{x-x_{1}}=1\)

We also make use of the fact that if two lines with gradients m1 and m2 respectively are perpendicular, then m1m2 = −1.

Example: Suppose we wish to find points on the curve y(x) given by \(x^{3} – 6x^{2} + x + 3 \) where the tangents are parallel to the line y = x + 5.

Solution: If the tangents have to be parallel to the line then they must have the same gradient. The standard equation for a straight line is y = mx + c, where m is the gradient. So what we gain from looking at this standard equation and comparing it with the straight line y = x + 5 is that the gradient, m, is equal to 1. Thus the gradients of the tangents we are trying to find must also have gradient 1.

We know that if we differentiate y(x) we will obtain an expression for the gradients of the tangents to y(x) and we can set this equal to 1. Differentiating, and setting this equal to 1 we find

\(\frac{dy}{xy}=3x^{2}-12x+1=1\)

from which

\(3x^{2} – 12x = 0\)

This is a quadratic equation which we can solve by factorisation

\(3x^{2} – 12x = 0\)

\(3x (x-4) = 0\)

\(\Rightarrow 3x = 0 \;\; or \;\; (x-4) = 0\)

\(\Rightarrow x = 0 \;\; or x = 4\)

Now having found these two values of x we can calculate the corresponding y coordinates. We do this from the equation of the curve: \(y = x^{3} – 6x^{2} + x + 3 \).

when x = 0: y = \((0)^{3} – 6(0)^{2} + 0 + 3 = 3\)

when x = 4: y = \((4)^{3} – 6(4)^{2} + 4 + 3= -25 \)

So the two points are (0, 3) and (4, −25)

These are the two points where the gradients of the tangent are equal to 1, and so where the tangents are parallel to the line that we started out with, i.e. y = x + 5.

#### 2. Secant lines

A line in the plane is a secant line to a circle if it meets the circle in exactly two points. It is also equivalent to the average rate of change, or simply the slope between two points. The average rate of change of a function between two points and the slope between two points are the same thing.

In the left figure above,

\(0=\frac{1}{2}\) (arc A C + arc B D),

while in the right figure,

\(\phi =\frac{1}{2}\) (arc R T – arc S Q),

Where arc AB denotes the angular measure of the arc AB