# 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

*x*-coordinate (

*always*comes first. The first number (the first coordinate) is

*always*on 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. The

*x*-coordinate is 2. The*y*-coordinate is 5. To locate (2,5), move 2 units to the right on the*x*-axis and 5 units up on the*y*-axis.
The order in which you write

*x*- and*y*-coordinates in an ordered pair is very important. Th e*x*-coordinate always comes first, followed by the*y*-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 (𝑥

_{1}, 𝑦_{1)}and (𝐵 𝑥_{2}, 𝑦_{2)}, the slope between the two points is given by:
Example 1

Find the gradient of the lines joining:

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

**Solution**

Example 2

- The line joining (2,−3) and (𝑘, 5) has gradient −2. Find 𝑘
- 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

- Gradient 2 and 𝑦 − intercept −4
- Gradient −2⁄3and passing through the point (2, 4)
- 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

- 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 and
- 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

- 2𝑦 = 5𝑥 + 1
- 2𝑥 + 3𝑦 = 5
- 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

- to get 𝑥 − intercept, let 𝑦 = 0 and
- 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:

- By using intercepts
- 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**

- Perimeter = 7𝑚 + 7𝑚 + 3𝑚 + 3𝑚 = 20 𝑚
- Perimeter = 2𝑚 + 4𝑚 + 5𝑚 = 11 𝑚
- 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 as

**3.14159**. 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

- diameter 9 𝑐𝑚
- radius 3½𝑚
- diameter 4.5 𝑑𝑚
- 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 𝑚

*, height, ℎ = 10 𝑚, one parallel side, 𝑎 = 4 𝑚. Let other parallel side be, 𝑏*^{2}
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**

**FORM ONE MATHEMATICS STUDY NOTES****FORM TWO MATHEMATICS STUDY NOTES****FORM THREE MATHEMATICS STUDY NOTES****FORM FOUR MATHEMATICS STUDY NOTES**
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