STRAND 4.0 GEOMETRY

1. STRAND 4.0:
GEOMETRY

2. SPECIFIC LEARNING OUTCOMES
By the end of the sub- strand, the learner
should be able to;
1. Construct parallel and perpendicular
lines in different situations
2. Divide a line proportionally in different
situations
3. Identify angle properties of polygons in
different situations

3. 4. Construct regular polygons up to a hexagon
in different situations
5. Construct irregular polygons up to a hexagon
in different situations
6. Construct circles passing through the vertices
of a triangle in different situations
7. Construct circles touching the sides of the
triangle in different situations
8. Admire geometric patterns in objects and
substances in real life.

4. PARALLEL LINES
Using a compass, draw an arc with the same
radius from two points on the line segment.
Without changing the radius of the compass,
draw arcs from the two points where the first
arc intersected the line segment.
The two lines that intersect the two arcs are
parallel to the original line segment.

5. ASSESSMENT
Construct a parallel to line JA passing through point C

6. CONSTRUCT PERPENDICULAR LINES
Draw a line segment.
Using a compass, draw an arc with any radius from one point on the line
segment.
Without changing the radius of the compass, draw an arc from the other point on
the line segment that intersects the first arc.
Draw a line through the point where the two arcs intersect and the point on the
line segment where the arc started.
The line that you just drew is perpendicular to the original line segment.

7. DIVIDING A LINE PROPORTIONALLY
Here are the steps on how to divide a line proportionally:
Draw a line segment.
Choose the number of parts you want to divide the line
into.
Using a compass, draw arcs with the same radius from two
points on the line segment, such that the arcs intersect
each other.
Draw lines from the points where the arcs intersect to the
two ends of the line segment.
The line segments that you just drew divide the original

8. ANGLE PROPERTIES OF
POLYGONS IN DIFFERENT
SITUATIONS
Triangle: The sum of the interior angles of a triangle
is always 180°. This is because any three points in a
plane can be connected to form a triangle, and the
sum of the angles at any point is always 180°.
Quadrilateral: The sum of the interior angles of a
quadrilateral is always 360°. This is because any four
points in a plane can be connected to form a
quadrilateral, and the sum of the angles at any point
is always 180°.

9. ANGLE PROPERTIES OF POLYGONS IN
DIFFERENT SITUATIONS
Pentagon: The sum of the interior angles of a
pentagon is always 540°. This is because any five
points in a plane can be connected to form a
pentagon, and the sum of the angles at any point is
always 180°.
Hexagon: The sum of the interior angles of a
hexagon is always 720°. This is because any six
points in a plane can be connected to form a
hexagon, and the sum of the angles at any point is
always 180°.

10. ANGLE PROPERTIES OF
POLYGONS
The exterior angles of a polygon add up
to 360°.
In a regular polygon, all the interior
angles are equal.
In a regular polygon, all the exterior
angles are equal.

11. CONSTRUCTING REGULAR POLYGONS
What is a polygon?
A polygon is a closed two-dimensional shape that is
formed by enclosing line segments. A minimum of three
line segments are required to make a polygon.
Hexagon- 6 sides
Pentagon—5 sides
Quadrilateral- 4 sides
Triangle- 3 sides

12. HOW TO CONSTRUCT A PENTAGON
USING A COMPASS AND A RULER
Draw a line segment of any length. This will be the base of
the pentagon.
Using the compass, draw an arc with a radius that is
slightly larger than half the length of the base. The center
of the arc should be on the base.
Without changing the radius of the compass, draw five
arcs that intersect the first arc.
Draw lines connecting the points where the arcs intersect.
These lines will form the sides of the pentagon.

13. CONSTRUCTING A PENTAGON
Make sure that the five arcs intersect each other at
the correct angle. The angle between two adjacent
sides of a pentagon is always 108°.
If you are constructing a regular pentagon, make
sure that all of the sides are the same in length.
You can use a protractor to help you measure the
angles between adjacent sides of the pentagon

14. HEXAGON
here are the steps on how to construct a regular hexagon using a compass and a ruler:
Draw a line segment of any length. This will be the base of the hexagon.
Using the compass, draw an arc with a radius that is slightly larger than half the length
of the base. The center of the arc should be on the base.
Without changing the radius of the compass, draw six arcs that intersect the first arc.
Draw lines connecting the points where the arcs intersect. These lines will form the
sides of the hexagon.

15. HERE ARE SOME ADDITIONAL TIPS FOR
CONSTRUCTING A HEXAGON
Make sure that the six arcs intersect each other at the correct angle.
The angle between two adjacent sides of a regular hexagon is always
120°.
If you are constructing a regular hexagon, make sure that all of the
sides are the same length
You can use a protractor to help you measure the angles between
adjacent sides of the hexagon

16. CONSTRUCTING AN IRREGULAR
POLYGON
Irregular Polygon: In case all the sides and
the interior angles of the polygon do not
measure similarly, then it is called an
irregular polygon. Examples of irregular
polygons include a rectangle, a kite

17. START WITH A BASE LINE
Draw a horizontal baseline to serve as a starting point for your polygon. This baseline
can be the bottom of your paper or any other reference line. Choose a Starting Point:
On the baseline, mark a point to begin your polygon. This will be the first vertex of
your polygon.
Determine the number of Sides: Decide on the number of sides for your polygon. Let's
construct a hexagon, which has six sides.
:

18. START WITH A BASE LINE
Measure Angles: Using a protractor, measure the angles you want for your polygon's
vertices. Since it's an irregular polygon, angles can vary. For a hexagon, you could
choose angles between 60 to 150 degrees.
Mark and Connect Vertices: From your starting point, measure and mark the first
angle along the baseline.
Using your ruler, draw a line from the starting point to the marked angle. This will be
your first side.
Measure and mark the next angle, and draw the next side from the endpoint of the
previous side.
Continue this process until you have drawn all six sides for the hexagon.
Check for Closure

19. START WITH A BASE LINE
Make sure the last side you draw connects back to
the starting point, closing the shape. Adjust angles
or side lengths if needed to ensure closure.
Label the Polygon:
You can label the angles and sides of your polygon
using letters or numbers if needed.
Note: Irregular polygons can have different shapes
and sizes

20. CONSTRUCT A CIRCLE TOUCHING VERTICES
OF A TRIANGLE
Here are the steps on how to construct circles passing through the vertices of a
triangle using a compass:
Draw a triangle.
Using the compass, draw an arc with any radius from one vertex of the triangle.
Without changing the radius of the compass, draw an arc from the other two vertices
of the triangle that intersect the first arc.
The points where the two arcs intersect are the centers of the circles that pass
through the vertices of the triangle.

21. CONSTRUCTING CIRCLES PASSING
THROUGH THE VERTICES OF A
TRIANGLE
Make sure that the two arcs intersect each other at the
correct angle. The angle between two adjacent sides of a
triangle is always 180°.
If you are constructing a circle with a specific radius, make
sure to adjust the radius of the compass accordingly.
You can use a protractor to help you measure the angles
between adjacent sides of the triangle

22. CONSTRUCT A CIRCLE THAT
TOUCHES THE SIDES OF A
TRIANGLE
Draw a triangle.
Choose any point on one of the sides of the triangle. This will be the center of the
circle.
Using the compass, draw an arc with any radius from the chosen point.
The points where the arc intersects the other two sides of the triangle are the
points where the circle touches the sides of the triangle.

23. CONSTRUCTING A CIRCLE THAT
TOUCHES THE SIDES OF A
TRIANGLE
Make sure that the arc intersects the other two sides of the triangle at the correct
angle. The angle between a tangent and a radius is always 90°.
If you are constructing a circle with a specific radius, make sure to adjust the
radius of the compass accordingly.
You can use a protractor to help you measure the angles between the arc and the
other two sides of the triangle.

24. APPLICATIONS OF POLYGONS IN REAL
LIFE
Engineering: Polygons are also used in engineering to create bridges, buildings, and
other structures
Architecture: Polygons are used in architecture to create strong and stable
structures. For example, the pyramids
Mathematics: Polygons are an important part of mathematics. They are used to study
shapes, angles, and symmetry.

25. SPECIFIC LEARNING OUTCOMES
By the end of the sub- strand, the learner should be able to;
draw a labelled Cartesian plane on different learning materials
identify points on the Cartesian plane in different situations
plot points on the Cartesian plane in different situations
generate table of values for a linear equation in different situations
determine an appropriate scale for a linear equation on the Cartesian plane in
different situations
draw a linear graph from table of values on Cartesian plane in different situations

26. SPECIFIC LEARNING OUTCOMES
solve simultaneous linear equations
graphically in different situations
apply simultaneous equations in real life
situations
use IT or other resources to learn more on
coordinates and graphs
reflect on the use of graphs in real life.

27. COORDINATES AND GRAPHS
Cartesian plane
A Cartesian plane is a plot of X and y axis .X representing Horizontal line and Y
representing vertical line
Coordinates are a pair of numbers that uniquely define the position of a point on a
plane. The first number is the x-coordinate, which tells you how far the point is to
the right of the origin.

28. PLOTTING IN THE CARTESIAN PLANE
Coordinates for the rectangle are
W(-3, 2), X(-3, 5), Y(4,5), Z(4,2)

29. TABLES FOR LINEAR EQUATION
A linear equation is an equation of the form y
= mx + b, where m and b are constants. The
variable x represents the independent variable,
and the variable y represents the dependent
variable.
Example 1: y = 2x + 3
X Y
0 3
1 5
2 7
3 9
4 11

30. CHOOSING A SCALE FOR YOUR
CARTESIAN PLANE
You should choose a scale that is large enough to
show the entire graph of the equation, but not so large
that the graph is too spread out and difficult to see.
You also want to choose a scale that is consistent with
the range of the values of x and y in the equation.
You can also choose different scales for your graph

31. DRAW A LINEAR GRAPH
from the graph above we get to see that Y-axis uses a scale of at least 10 in a
range units while in the X axis, it uses a range of 1 unit
solving simultaneous equations graphically
Equation 1: y = 6-x
Equation 2: y = 2+3x
Solve both equations
Equation 1
X -3 -2 -1 0 1 2 3 4
y 9 8 7 6 5 4 3 2

32. APPLICATION OF SIMULTANEOUS
EQUATION
Simultaneous equations can be applied to a wide variety of real-life situations.
Here are a few examples
Budgeting: You can use simultaneous equations to create a budget that meets all
of your financial goals. For example, you might want to create a budget that
allows you to save money for a down payment on a house, while also paying off
debt and having enough money for food and other expenses
Investing: You can use simultaneous equations to decide how to allocate your
investment money between different businesses.
Choosing the best deal: You can use simultaneous equations to compare
different deals and find the best one for you. For example, you might be
comparing different car insurance policies.

33. APPLICATION OF GRAPHS
Graphs are a powerful tool that can be used to represent and analyze data in a variety
of ways. They can be used to visualize trends, identify patterns, and make
predictions. Graphs can be used in a wide variety of real-life situations, including:
A company might use a linear graph to track its sales over time. This information
could be used to identify trends, such as seasonal fluctuations or changes in
marketing campaigns. The company could then use this information to make
decisions about its pricing, inventory, or advertising.
A scientist might use a linear graph to track the growth of bacteria over time. This
information could be used to determine the optimal growth conditions for the
bacteria, or to identify the effects of different treatments.

34. APPLICATION OF GRAPHS
An engineer might use a linear graph to design a bridge. The graph would show
the relationship between the weight of the bridge and the amount of force it can
withstand. This information would be used to ensure that the bridge is strong
enough to support its load.
A financial advisor might use a linear graph to track the stock market. This
information could be used to identify undervalued stocks, or to predict future
trends.
A teacher might use a linear graph to teach students about relationships between
different quantities. For example, the teacher could use a graph to show how the
cost of a car changes depending on its fuel efficiency.

35. QUIZ
Solve the simultaneous equation
graphically
a) y=3x-1 b) 5x+y=7
2y+2x=3
3x+2y=0
2.For each of the following linear
equation complete the table and draw

36. QUIZ
Draw a graph for the following equations
a) 3x-y=3 b)𝑦
2+2x =5
For this following equation, rewrite in the form of y= mx+b and draw their graphs
a) 3y+4x =6
6y = 1—8x
ABCD is a rectangle. If A (0,0), B(4,0), C(4,6) are three vertices of a rectangle, find
the coordinates of D fourth vertex

37. SPECIFIC LEARNING OUTCOMES
By the end of the sub- strand, the learner should be able to;
represent length to a given scale in different situations.
convert actual length to scale length in real life situations
convert scale length to actual length in real life situations.
interpret linear scales in statement form in different situations.
write linear scales in statement form in different situations.
interpret linear scales in ratio form in different situations.
write linear scales in ratio form in different situations.
convert linear scale from statement form to ratio form and ratio form to statement
form in different situations

38. SCALE DRAWING
Representing length to a given scale is a
common practice in various situations,
especially in maps. The scale is a ratio
that shows the relationship between the
length on the drawing and the actual
length in real life. Here's how to
represent length to a given scale in
different situations

39. REPRESENTING LENGTH ON A MAP
Suppose you have a map of a city, and the
scale is 1 cm = 2 km. If you want to represent
a road that is 6 km long on the map, follow
these steps:
1. Convert the actual length to the map scale:
6 km × (1 cm/2 km) = 3 cm
2. On the map, draw a line that is 3 cm long
to represent the road.

40. REPRESENTING LENGTH ON A
BLUEPRINT
In a blueprint for a furniture design, the scale
might be 1 cm = 10 cm. If you want to represent
the length of a table that is 120 cm long, follow
these steps:
1. Convert the actual length to the map scale: 120
cm × (1𝑐𝑚
10𝑐𝑚) = 12 cm
2. On the blueprint, draw a line that is 12 cm long
to represent the table.

41. CONVERTING ACTUAL LENGTH TO
SCALE LENGTH
The scale of a map is given as
1:250,000.write this as a statement
1:250,000 means 1 cm in the map
represents 250,000cm on the ground.
Therefore, 1cm represents 250000
100000km
i.e 1cm represents 2.5km

42. CONVERTING SCALE LENGTH TO
ACTUAL LENGTH
For example, if the scale of a scale
model is 1:100 and the actual length
of a car is 5 m, then the
corresponding length of the car on
the scale model is 0.05 m.
1
100 × 5 =0.05

43. IN A STATEMENT FORM, A LINEAR
SCALE CAN BE INTERPRETED AS
FOLLOWS
If one variable increases, the other variable also increases. This is called a positive
relationship. For example, if you plot the height of a plant over time, you will see that
the plant gets taller as time goes on. This is a positive relationship because the height
of the plant is increasing and the time is increasing.
If one variable increases, the other variable decreases. This is called a negative
relationship. For example, if you plot the temperature outside over time, you will see
that the temperature gets colder as time goes on. This is a negative relationship
because the temperature is decreasing and the time is increasing.
The two variables are not related to each other. This is called a zero relationship. For
example, if you plot the number of students in a class over time, you will see that the
number of students does not change. This is a zero relationship because the number
of students is not related to time

44. INTERPRETING LINEAR SCALES IN
RATIO FORM
i) In a scale diagram 1cm represents 50 m
(a) Write the scale in ratio form.
1cm represents 50m =50× 100𝑐𝑚 = 5000𝑐𝑚
Scale in ratio form = 1:5000
Find the drawing length for 1.25 km.
1.25km× 1000𝑚 = 1250𝑚
1250
50 = 25𝑚

45. EXAMPLE
Find the actual length in kilometers corresponding to a length of 10.5cm on the diagram.
10.5 cm represents 10.5 × 50cm=525m=0.525km
ii) In a diagram of a plan of a factory a length of 2.5 cm represents an actual length of 12.5 m
(a) Work out the linear scale in ratio form.
2.5cm represents 12.5m
12.5× 100 = 1250
2.5:1250
1250
2.5 =500cm
𝑟𝑎𝑡𝑖𝑜 = 1: 500𝑐𝑚

46. RATIO TO STATEMENT AND STATEMENT TO
RATIO
Write each of the following scales as a ratio
(i)1cm represents 4km
4× 100000 = 400000
1:400000
(ii)1cm represents 5m
5× 100 = 500𝑐𝑚
1:500
(iii)1cm represents 200m
200× 100 = 20000𝑐𝑚

47. RATIO TO STATEMENT
1cm:200m- this shows that for every 1cm in the scale corresponds to
200m actual length
2.1cm:4km- this shows that for every 1cm in the scale corresponds
to 4km actual distance

48. MAKING SCALE DRAWINGS
-To make a scale drawing we need to choose a linear scale.
The scale is 1:25. The height of the main building of the White house is 85 feet. Find
this height on the model.
Solution
1
25=ℎ
85
Write the proportion to find the height (h) of the model.
1 × 85 = 25 h
cross multiply to solve 85
25 = 25ℎ
25
Divide both sides by 25.
3.4 = h
Answer: The height of the main building of the model is 3.4 feet

49. APPLICATIONS OF SCALING
Scale drawings are used in many real-life situations, such as:
Engineering: Engineers use scale drawings to design and plan
structures, such as buildings, bridges, and roads. Scale
drawings allow engineers to visualize the final product and to
make sure that it is safe and efficient.
Architecture: Architects use scale drawings to create detailed
plans for buildings. Scale drawings allow architects to
communicate their ideas to clients and to contractors.
Construction: Contractors use scale drawings to build
structures according to the plans created by engineers and
architects. Scale drawings allow contractors to accurately
measure and cut materials, and to ensure that the structure is

50. USES OF SCALING
I appreciate the use of scale drawings in maps too. They are a very
important tool for visualizing and understanding the world around us.
Without scale drawings, maps would be much less useful.
Here are some of the reasons why I appreciate the use of scale drawings
in maps:
They allow us to see the big picture. Maps can show us large areas of
land, such as countries or continents. This can be helpful for
understanding the relationships between different places.
They are accurate. Scale drawings are based on real measurements, so
they are accurate representations of the real world. This is important for
making sure that we can trust the information that we are seeing on a
map.
They are easy to use. Scale drawings are relatively easy to understand,

51. QUIZ
On a map whose scale is 1:50000, a rectangular piece of land measures 3 cm by 2 cm.
What is the size of this piece of land in hectares?
The scale below represents three towns X, Y, and Z. The distance from town Y to town
Z is 550 km
What is the distance from town Y to town Z through town X?
The distance between town P and town Q on a map is 5 cm and the actual distance is
5 km, what is the scale in the map?
A rectangular field measuring 420 m by 600 m is to be represented on a scale
drawing using the scale 1:20000. What is the area of the scale drawing in cm2
A length of 3.5 cm on a map represents a distance of 1.75 km on the ground.
a) Find the scale of the map in ratio form.
b) Calculate the length on the map on a road which is 12.5 km.

52. SPECIFIC LEARNING OUTCOMES
By the end of the sub- strand, the learner should be
able to;
identify common solids from the environment
sketch nets of cubes, cuboids, cylinders, pyramids and
cones in different situations
work out surface area of the solids from nets of solids
in different situations
determine the distance between two points on the
surface of a solid in different situations

53. COMMON SOLIDS
Identify common solids
Cubes: A cube is a six-sided solid with all sides (faces) being
equal squares. Examples include dice and Rubik's cubes.
Spheres: Spheres are perfectly round solids with all points on
the surface equidistant from the center. Examples include balls
and oranges.
Cylinders: Cylinders have two circular bases that are parallel
and congruent. Examples include cans and drinking glasses.
Cones: Cones have a circular base and a curved surface that
tapers to a point called the apex. Examples include ice cream
cones and traffic cones.
Pyramids: Pyramids have a polygon base and triangular faces

54. SKETCH OF COMMON SOLIDS USING
NETS
The first solid is a cube
Cylinder
Cone
Pyramid

55. SURFACE AREA OF A CUBE
To find the surface area of a cube we must count the
number of surface faces and add the areas of each of them
together.
In a cube there are 6 faces, each a square with the same
side lengths.
In this example the side lengths is 15 cmso the area of
each square would be 152=225
We then multiply this number by 6, the number of faces of
the cube, to get 225×6=1350
Our answer for the surface area is 1350cm.
3

56. SURFACE AREA OF A CUBOID
SA=2( lw+𝑤ℎ + 𝑙ℎ) where l=length, w = width,
h =height
A cuboid measures 30 cm by 10 cm by 8 cm.
Calculate it's total surface area.
2(30× 10 + 10 × 8 + 30 × 8)
2(620)cm2
=1240cm2
Volume of a cuboid: L × 𝑤 × ℎ

57. SURFACE AREA OF A CYLINDER
The general formula for the total
surface area of a cylinder
Is T.S.A=2𝜋𝑟2+2𝜋𝑟ℎ
Volume of a cylinder: 𝜋𝑟2h where
h=height

58. SURFACE AREA OF A CONE
Volume of a pyramid:𝑙×𝑤×ℎ
3
Distance between two points in a solid
Find the distance between BX
BX=BG+GF+FX
=4+3+ √ ( 22+ 52)
=7+√29
=7+5.385
=12.385

59. APPLICATION OF COMMON
SOLIDS
In engineering: Common solids are used to design machines, engines, and other
devices
In manufacturing: Common solids are used to create products, such as toys, tools,
and furniture. For example, cuboids are often used to create boxes, while pyramids
are often used to create dice.
In education: Common solids are used to teach mathematics, science, and
engineering concepts. For example, cubes are often used to teach volume, while
spheres are often used to teach surface area.
In architecture : Common solids are used to design buildings, bridges, and other
structures. For example, cubes are often used to create simple, symmetrical
structures, while cylinders are often used to create curved, flowing structures.

60. QUIZ
. A house which has a cement flat roof top measures 18m
long, 10m wide and 8m high. Find the surface area of the
walls without the roof.
2.Use net to calculate the surface area of a cylinder whose
radius and height are 7cm and 10cm respectively
3.Calculate the surface area of a pyramid whose base is
square with length is 10cm and slant side 15cm long
4. What is the volume of the cylinder below with a radius =
7cm and height 31.5cm?

61. QUIZ
4. What is the volume of the cylinder below with a radius = 7cm and
height 31.5cm?
31.5cm

62. SPECIFIC LEARNING OUTCOMES
By the end of the sub- strand, the learner should be able to;
interpret bar graphs of data from real life situations
Draw bar graphs of data from real life situations
Draw line graphs of given data from real life situations
Interpret line graphs of data from real life situations
Identify the mode of a set of discrete data from real life situations
Calculate the mean of a set of discrete data from real life situations
Determine the median of a set of discrete data from real life situations
Use IT or other materials to determine the mean, mode and median of discrete
data in different situations.
Recognize use of data representation and interpretation in real life situations.