**TOPIC 1: COORDINATE GEOMETRY ~ MATHEMATICS FORM 4**

**Equation of a Line**

The General Equation of a Straight Line

Derive the general equation of a straight line

**COORDINATES OF A POINT**

•The coordinates of a points – are the values of *x* and* y enclosed by the brackets which are used to describe the position of point in a line in the plane.*

The plane is called *xy*-plane and it has two axis.

- horizontal axis known as axis and
- vertical axis known as axis

*xy*-plane below

- (3,6) and (-2,8)
- (0,6) and (99,-12)
- (4,5)and (5,4)

2. A line passes through (3, a) and (4, -2), what is the value of a if the slope of the line is 4?

**FINDING THE EQUATION OF A STRAIGHT LINE**

The equation of a straight line can be determined if one of the following is given:-

- Gradient 2 and intercept
- Gradient and passing through the point
- Passing through the points and

**Solution**

**EQUATION OF A STRAIGHT LINE IN DIFFERENT FORMS**

**INTERCEPTS**

**Attempt the following Questions.**

Find the y-intercept of the line 3x+2y = 18 .

What is the x-intercept of the line passing through (3,3) and (-4,9)?

Calculate the slope of the line given by the equation x-3y= 9

Find the equation of the straight line with a slope -4 and passing through the point (0,0).

Find the equation of the straight line with y-intercept 5 and passing through the point (-4,8).

**GRAPHS OF STRAIGHT LINES**

- By using intercepts
- By using the table of values

**SOLVING SIMULTANEOUS EQUATION BY GRAPHICAL METHOD**

Use the intercepts to plot the straight lines of the simultaneous equations

The point where the two lines cross each other is the **solution** to the simultaneous equations

**Exercise 3**

1. Draw the line 4x-2y=7 and 3x+y=7 on the same axis and hence determine their intersection point

- y-x = 3 and 2x+y = 9
- 3x- 4y=-1 and x+y = 2
- x = 8 and 2x-3y = 10

**Midpoint of a Line Segment**

The Coordinates of the Midpoint of a Line Segment

_{1},y

_{1}), T with coordinates (x

_{2},y

_{2}) and M with coordinates (x,y) where M is the mid-point of ST. Consider the figure below:

Therefore the coordinates of the midpoint of the line joining the points (-2,8) and (-4, -2) is (-3,3).

**Distance Between Two Points on a Plane**

Consider two points, A(x_{1},y_{1}) and B(x_{2},y_{2}) as shown in the figure below:

The distance between A and B in terms of x_{1}, y_{1,}x_{2}, and y_{2}can be found as follows:Join AB and draw doted lines as shown in the figure above.

_{2}– x

_{1}and BC = y

_{2}– y

_{1}

Therefore the distance is 13 units.

**Parallel and Perpendicular Lines**

The two lines which never meet when produced infinitely are called parallel lines. See figure below:

The two parallel lines must have the same slope. That is, if M_{1}is the slope for L_{1}and M_{2}is the slope for L_{2}thenM_{1}= M_{2}

When two straight lines intersect at right angle, we say that the lines are perpendicular lines. See an illustration below.

_{1}(x

_{1},y

_{1}), P

_{2}(x

_{2},y

_{2}), P

_{3}(x

_{3},y

_{3}), R(x

_{1},y

_{2}) and Q(x

_{3},y

_{2}) and the anglesα,β,γ(alpha, beta and gamma respectively).

- α+β = 90 (complementary angles)
- α+γ= 90 (complementary angles)
- β = γ (alternate interior angles)

_{2}QP

_{3}is similar to triangle P

_{1}RP

_{2.}

Generally two perpendicular lines L_{1}and L_{2}with slopes M_{1}and M_{2}respectively the product of their slopes is equal to negative one. That is M_{1}M_{2}= -1.

Example 10

Let us find the slope of the lines AB, DC, AD and BC

We

see that each two opposite sides of the parallelogram have equal slope.

This means that the two opposite sides are parallel to each other,

which is the distinctive feature of the parallelogram. Therefore the

given vertices are the vertices of a parallelogram.

Example 11

angled triangle has two sides that are perpendicular, they form 90°.We

know that the slope of the line is given by: slope = change in y/change

in x

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