### General

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### General

### UNIT 13 : Points, straight lines and circles in 2D

**Key unit competence**Determine algebraic representations of lines, straight lines and circles in 2D.

**Learning objectives****13.1 Points in 2D****Activity 13.1**What is a point?

In Junior Secondary, we studied points on a Cartesian plane. How do we define the coordinate of a point in 2D? Discuss in groups and present your findings to the rest of the class

A point is an exact position or location on a plane surface. It is important to understand that a point is not a thing, but a place. We indicate the position of a point by placing a dot with a pencil. In coordinate geometry, points are located on the plane using their coordinates - two numbers that show where the point is positioned with respect to two number line “axes” at right angles to each other.

**Cartesian coordinate of a point****Distance between two points****The mid-point of a line segment****13.2 Lines in 2D****The equations of straight lines**A particular line is uniquely located in a plane if

• it has a known direction and passes through a known fixed point, or

• it passes though two known points.

**Cartesian equation of a straight line****Equation of a line given a law**We can find the equation of the locus by considering a point P(x, y) on the locus and using the law to derive an equation in x and y. This will be the equation of the locus.

**Parallel and perpendicular lines**If two lines are parallel, they have equal gradients

If two lines are perpendicular, the product of their gradients is –1.

**Vector, parametric, scalar and Cartesian equations of the line**In 2D, a line can be defined by an equation in slope–intercept form, a vector equation, parametric equations, or a Cartesian equation (scalar equation).

**Intersection of lines with equations in Cartesian form**Any point on a line has coordinates which will satisfy the equation of that line. In order to find the point in which two lines intersect we have to find a point with coordinates which satisfy both equations. This is equivalent, from an algebraic point of view, to solving the equations of the lines simultaneously.

**The intersection of two lines with equations given in vector form**In order to find the point of intersection of two lines whose equations are given in vector form each equation must have a separate parameter. The method as illustrated in the following example:

**13.3 Points and lines****Distance of a point from a line****Angle between two straight lines****13.4 The circle****Mental task**What is a circle? What are the main parts of a circle that you can recall? Why are they significant?

In fact

**the definition**of a circle is: the set of all points on a plane that are at a fixed distance from a centre.**Unit circle**If we place the circle centre at (0,0) and set the radius to 1 we get:

**Intersecting a line and a circle**Consider a straight line y = mx + c and a circle (x – a)

^{2}+ (y – b)^{2}= r^{2}.

There are three possible situations:

1. The line cuts the circle in two distinct places, i.e. part of the line is a chord of the circle.

2. The line touches the circle, i.e. the line is a tangent to the circle.

3. The line neither cuts nor touches the circle.

**Circle through three given points**Three non-collinear points define a circle,

i.e. there is one, and o nly one circle which can be drawn through three non-collinear points. The equation of any

**Supplementary, interactive questions served by Siyavula Education.**Note: Questions will open in a new window or tab.

**Points in 2D**- Reading a point from a graph
- Reading a point from a graph
- The distance formula: substituting correctly
- Calculating a line length from two points
- Getting to know the distance formula
- Finding a coordinate from a distance
- The distance formula with surds
- Finding the mid-point of two points
- Finding the mid-point from coordinates
- Mid-point calculations with algebra
- Using a mid-point to find an end point
- Working with algebraic coordinates and the mid-point

**Supplementary, interactive questions served by Siyavula Education.**Note: Questions will open in a new window or tab.

**Lines in 2D**- Simultaneous equations: the elimination method
- Simultaneous equations: the substitution method
- Simultaneous equations
- Simultaneous equations on the Cartesian plane
- Simultaneous equations
- Picking a solution method
- Solving simultaneous equations by elimination
- Solving simultaneous equations by substitution
- Simultaneous equations concepts
- Setting up elimination: picking a multiple
- Solving simultaneous equations
- Solving simultaneous equations by elimination
- Equations with fractions
- Solving simultaneous equations by substitution
- Finding equations of perpendicular lines

**Supplementary, interactive questions served by Siyavula Education.**Note: Questions will open in a new window or tab.

**The circle**- Finding the equation of a circle centred on the origin
- Equation of a circle
- Circle equations
- Equation of a circle: two points given
- Points on a circle: finding the missing coordinates
- Using a graph to determine the equation of the circle
- Equation of a circle
- Investigating the equations of a circle
- Circle equations and trigonometry
- Word problem: equation of a circle with centre at the origin
- Determining the equation of a circle
- Finding the equation of a circle with centre (a;b)
- Equation of a circle: completing the square
- Circle equation
- Find the coordinates