Saturday, January 21, 2023

MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY
Saturday, January 21, 2023
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.
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

Equation of a Line

The General Equation of a Straight LineCOORDINATES 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

Consider the xy-plane below

The coordinates of points A, B, C ,D and E are A(2, 3), B(4, 4), C(-3, -1), D(2, -4) and E(1, 0).

Definition
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY


Example 1

Find the gradient of the lines joining
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY


Example 2

(a) The line joining (2, -3) and (k, 5) has a gradient -2. Find k
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

Exercise 1

1. Find the gradientof the line which passes through the following points ;
(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?

3. The gradient of the linewhich goes through (4,3) and (-5,k) is 2. Find the value of k.

FINDING THE EQUATION OF A STRAIGHT LINE

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

MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY


Example 3

Find the equation of the line with the following

Gradient 2 and intercept
Gradient and passing through the point
Passing through the points and
Solution
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

EQUATION OF A STRAIGHT LINE IN DIFFERENT FORMS


The equation of a line can be expressed in two forms
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY


Example 4

Find the gradient of the following lines
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

INTERCEPTS
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

Therefore
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY


Example 5

Find the y-intercept of the following lines
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY


Example 6

Find the x and y-intercept of the following lines
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

Exercise 2

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

The graph of straight line can be drawn by using the following methods;
By using intercepts
By using the table of values


Example 7

Sketch the graph of Y = 2X - 1
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

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


Example 8

Solve the following simultaneous equations by graphical method
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY


Exercise 3

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

2. Find the solutionfor each pair the following simultaneous equations by graphical method;
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
Let S be a point with coordinates (x1,y1), T with coordinates (x2,y2) and M with coordinates (x,y) where M is the mid-point of ST. Consider the figure below:
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

Considering the angles of the triangles SMC and TMD, the triangles SMC and TMD are similar since their equiangular
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY


Example 9

Find the coordinates of the mid-point joining the points (-2,8) and (-4,-2)
Solution
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

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

The Distance Between Two Points on a Plane

Consider two points, A(x1,y1) and B(x2,y2) as shown in the figure below:
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

The distance between A and B in terms of x1, y1,x2, and y2can be found as follows:Join AB and draw doted lines as shown in the figure above.

Then, AC = x2– x1and BC = y2– y1

Since the triangle ABC is a right angled, then by applying Pythagoras theorem to the triangle ABC we obtain
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

Therefore the distance is 13 units.

Parallel and Perpendicular Lines

Gradients in order to Determine the Conditions for any Two Lines to be Parallel
The two lines which never meet when produced infinitely are called parallel lines. See figure below:
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

The two parallel lines must have the same slope. That is, if M1is the slope for L1and M2is the slope for L2thenM1= M2

Gradients in order to Determine the Conditions for any Two Lines to be Perpendicular
When two straight lines intersect at right angle, we say that the lines are perpendicular lines. See an illustration below.
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

Consider the points P1(x1,y1), P2(x2,y2), P3(x3,y3), R(x1,y2) and Q(x3,y2) 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)

Therefore the triangle P2QP3is similar to triangle P1RP2.
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

Generally two perpendicular lines L1and L2with slopes M1and M2respectively the product of their slopes is equal to negative one. That is M1M2= -1.

Example 10

Show that A(-3,1), B(1,2), C(0,-1) and D(-4,-2) are vertices of a parallelogram.
Solution

Let us find the slope of the lines AB, DC, AD and BC
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

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.

Problems on Parallel and Perpendicular Lines

Example 11

Show that A(-3,2), B(5,6) and C(7,2) are vertices of a right angled triangle.
Solution

Right 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

Now,
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY

Since the slope of AB and BC are negative reciprocals, then the triangle ABC is a right angled triangle at B.
MATHEMATICS FORM FOUR TOPIC 1: COORDINATE GEOMETRY
4/ 5
Oleh

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