FORM ONE MATHEMATICS STUDY NOTES TOPIC 10-11.

TOPIC 10: COORDINATE OF A  POINT

Read the coordinates of a point

Coordinates of a points – are the values of 𝑥 and 𝑦 enclosed by the bracket which are used to describe the position of a point in the plane

The plane used is called 𝑥𝑦 − plane and it has two axis; horizontal axis known as 𝑥 − axis and; vertical axis known as 𝑦 − axis

A Point Given its Coordinates

Plot a point given its coordinates

Suppose you were told to locate (5, 2) on the plane. Where would you look? To understand the meaning of (5, 2), you have to know the following rule: Thex-coordinate (alwayscomes first. The first number (the first coordinate) isalwayson the horizontal axis.

A Point on the Coordinates

Locate a point on the coordinates

The location of (2,5) is shown on the coordinate grid below. Thex-coordinate is 2. They-coordinate is 5. To locate (2,5), move 2 units to the right on thex-axis and 5 units up on they-axis.

The order in which you writex- andy-coordinates in an ordered pair is very important. Th ex-coordinate always comes first, followed by they-coordinate. As you can see in the coordinate grid below, the ordered pairs (3,4) and (4,3) refer to two different points!

Gradient (Slope) of a Line

The Gradient of a Line Given Two Points

Calculate the gradient of a line given two points

Gradient or slope of a line – is defined as the measure of steepness of the line. When using coordinates, gradient is defined as change in 𝑦 to the change in 𝑥.

Consider two points 𝐴 (𝑥₁, 𝑦₁₎and (𝐵 𝑥₂, 𝑦₂₎, the slope between the two points is given by:

Example 1

Find the gradient of the lines joining:

1. (5, 1) and (2,−2)
2. (4,−2) and (−1, 0)
3. (−2,−3) and (−4,−7)

Solution

Example 2

1. The line joining (2,−3) and (𝑘, 5) has gradient −2. Find 𝑘
2. Find the value of 𝑚 if the line joining the points (−5,−3) and (6,𝑚) has a slope of½

Solution

Equation of a Line

The Equations of a Line Given the Coordinates of Two Points on a Line

Find the equations of a line given the coordinates of two points on a line

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

• The gradient and the 𝑦 − intercept (at x = 0) or 𝑥 − intercept ( at y=0)
• The gradient and a point on the line
• Since only one point is given, then

• Two points on the line

Example 3

Find the equation of the line with the following

1. Gradient 2 and 𝑦 − intercept −4
2. Gradient −2⁄3and passing through the point (2, 4)
3. Passing through the points (3, 4) and (4, 5)

Solution

Divide by the negative sign, (−), throughout the equation

∴The equation of the line is 2𝑥 + 3𝑦 − 16 = 0

The equation of a line can be expressed in two forms

1. 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 and
2. 𝑦 = 𝑚𝑥 + 𝑐

Consider the equation of the form 𝑦 = 𝑚𝑥 + 𝑐

𝑚 = Gradient of the line

Example 4

Find the gradient of the following lines

1. 2𝑦 = 5𝑥 + 1
2. 2𝑥 + 3𝑦 = 5
3. 𝑥 + 𝑦 = 3

Solution

Intercepts

The line of the form 𝑦 = 𝑚𝑥 + 𝑐, crosses the 𝑦 − 𝑎𝑥𝑖𝑠 when 𝑥 = 0 and also crosses 𝑥 − 𝑎𝑥𝑖𝑠 when 𝑦 = 0

See the figure below

Therefore

1. to get 𝑥 − intercept, let 𝑦 = 0 and
2. to get 𝑦 − intercept, let 𝑥 = 0

From the line, 𝑦 = 𝑚𝑥 + 𝑐

𝑦 − intercept, let 𝑥 = 0

𝑦 = 𝑚 0 + 𝑐 = 0 + 𝑐 = 𝑐

𝑦 − intercept = c

Therefore, in the equation of the form 𝑦 = 𝑚𝑥 + 𝑐, 𝑚 is the gradient and 𝑐 is the 𝑦 − intercept

Example 5

Find the 𝑦 − intercepts of the following lines

Solution

Graphs of Linear Equations

The Table of Value

Form the table of value

The graph of a straight line can be drawn by using two methods:

1. By using intercepts
2. By using the table of values

Example 6

Sketch the graph of 𝑦 = 2𝑥 − 1

Solution

The Graph of a Linear Equation

Draw the graph of a linear equation

By using the table of values

Simultaneous Equations

Linear Simultaneous Equations Graphically

Solve linear simultaneous equations graphically

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 7

Solve the following simultaneous equations by graphical method

Solution

Consider: 𝑥 + 𝑦 = 4

If 𝑥 = 0, 0 + 𝑦 = 4 𝑦 = 4

If 𝑦 = 0, 𝑥 + 0 = 4 𝑥 = 4

Draw a straight line through the points 0, 4 and 4, 0 on the 𝑥𝑦 − plane

Consider: 2𝑥 − 𝑦 = 2

If 𝑥 = 0, 0 − 𝑦 = 2 𝑦 = −2

If 𝑦 = 0, 2𝑥 − 0 = 2 𝑥 = 1

Draw a straight line through the points (0,−2) and (1, 0) on the 𝑥𝑦 − plane

From the graph above the two lines meet at the point 2, 2 , therefore 𝑥 = 2 𝑎𝑛𝑑 𝑦 = 2

TOPIC 11: PERIMETERS AND AREAS 

Perimeters of Triangles and Quadrilaterals

The Perimeters of Triangles and Quadrilaterals

Find the perimeters of triangles and quadrilaterals

Perimeter – is defined as the total length of a closed shape. It is obtained by adding the lengths of the sides inclosing the shape. Perimeter can be measured in 𝑚𝑚 , 𝑐𝑚 ,𝑑𝑚 ,𝑚,𝑘𝑚 e. t. c

Examples

Example 1

Find the perimeters of the following shapes

Solution

1. Perimeter = 7𝑚 + 7𝑚 + 3𝑚 + 3𝑚 = 20 𝑚
2. Perimeter = 2𝑚 + 4𝑚 + 5𝑚 = 11 𝑚
3. Perimeter = 3𝑐𝑚 + 6𝑐𝑚 + 4𝑐𝑚 + 5𝑐𝑚 + 5 𝑐𝑚 + 4𝑐𝑚 = 27 𝑐𝑚

Circumference of a Circle

The Value of Pi ( Π)

Estimate the value of Pi ( Π)

The number π is a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as3.14159. It has been represented by the Greek letter "π" since the mid 18th century, though it is also sometimes spelled out as "pi" (/paɪ/).

The perimeter of a circle is the length of its circumference 𝑖. 𝑒 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒. Experiments show that the ratio of the circumference to the diameter is the same for all circles

The Circumference of a Circle

Calculate the circumference of a circle

Example 2

Find the circumferences of the circles with the following measurements. Take 𝜋 = 3.14

1. diameter 9 𝑐𝑚
2. radius 3½𝑚
3. diameter 4.5 𝑑𝑚
4. radius 8 𝑘𝑚

Solution

Example 3

The circumference of a car wheel is 150 𝑐𝑚. What is the radius of the wheel?

Solution

Given circumference, 𝐶 = 150 𝑐𝑚

∴ The radius of the wheel is 23.89 𝑐𝑚

Areas of Rectangles and Triangles

The Area of a Rectangle

Calculate the area of a rectangle

Area – can be defined as the total surface covered by a shape. The shape can be rectangle, square, trapezium e. t. c. Area is measured in mm!, cm!,dm!,m! e. t. c

Consider a rectangle of length 𝑙 and width 𝑤

Consider a square of side 𝑙

Consider a triangle with a height, ℎ and a base, 𝑏

Areas of Trapezium and Parallelogram

The Area of a Parallelogram

Calculate area of a parallelogram

A parallelogram consists of two triangles inside. Consider the figure below:

The Area of a Trapezium

Calculate the area of a trapezium

Consider a trapezium of height, ℎ and parallel sides 𝑎 and 𝑏

Example 4

The area of a trapezium is120 𝑚!. Its height is 10 𝑚 and one of the parallel sides is 4 𝑚. What is the other parallel side?

Solution

Given area, 𝐴 = 120 𝑚², height, ℎ = 10 𝑚, one parallel side, 𝑎 = 4 𝑚. Let other parallel side be, 𝑏

Then

Area of a Circle

Areas of Circle

Calculate areas of circle

Consider a circle of radius r;

Example 5

Find the areas of the following figures

Solution

Example 6

A circle has a circumference of 30 𝑚. What is its area?

Solution

Given circumference, 𝐶 = 30 𝑚

C = 2𝜋𝑟

FORM ONE MATHEMATICS OTHER TOPICS
FORM ONE MATHEMATICS TOPIC 1-3.
FORM ONE MATHEMATICS STUDY NOTES TOPIC 4-6.
FORM ONE MATHEMATICS STUDY NOTES TOPIC 7-9.
FORM ONE MATHEMATICS STUDY NOTES TOPIC 10-11.

 

O'LEVEL MATHEMATICS NOTES 
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