STRAND 4 GEOMETRY.pptx

1. STRAND 4.0: GEOMETRY
Sub-Strand 3.0: Construct perpendicular and parallel lines

2. 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.

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

4. SPECIFIC LEARNING OUTCOMES
• construct regular polygons up to a hexagon in different situations
• construct irregular polygons up to a hexagon in different situations
• construct circles passing through the vertices of a triangle in
different situations
• construct circles touching the sides of the triangle in different
situations
• admire geometric patterns in objects and substances in real life.

5. 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.

6. 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 line segment
into the desired number of parts.

7. 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°.
• 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°.

8. 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.

9. 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

10. 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.

11. 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

12. 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.

13. 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

14. 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

15. 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.
• :

16. 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

17. 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

18. 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.

19. 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

20. 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.

21. 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.

22. 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.

23. 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

24. 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.

25. 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.

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

27. 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

28. 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

29. 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

30. 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.

31. 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.

32. 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.

33. 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 a graph
•a)y=6x+1
2

34. 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

35. 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

36. 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

37. 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.

38. 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.

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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 built correctly.

48. 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, even for
people who are not familiar with maps. This makes them a valuable tool for
people of all ages.

49. 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.

50. 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

51. 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 that meet
at a common vertex (apex). Examples include the Great Pyramid of Giza
and pyramid-shaped roofs on buildings.

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

53. 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.
• Volume of a cube: a3 where a is edge

54. 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

55. 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

56. 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

57. 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.

58. 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?

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