form 5

Chapter 12 : Number bases


12.1 : Numbers in Base Two , Eight, and Five

*Stating numbers in base two,eight and five

(a) Numbers in :

• Base two use only two digits , 0 , 1.
• Base eight use only eight digits,1,2,3,4,5,6 and 7.
• Base five use only five digits 0,1,2,3 and 4.

(b) Zero ,one, two, three, ... in base two , eight and five  are written as follows .




*Place value of a digit of a number in base two,eight and five are given in the table below.

a) The value of a digit of a number in base two,eight or five is determined as follows.

          

(b) Place value of a digit of a number in base two ,eight and five are given in table below.


*Writing numbers in base two, eight and five in expanded notation 

• A number in base two,eight or five can be written in expanded notation according to the corresponding place value of each digit.
• The 351710 in base ten can be written in expanded notation in the form of 3 x 10³ + 5 x 10² + 1 x 10¹ + 7 10 x 10°.

*Converting numbers in base two,eight and five to numbers in base ten and vice versa

(a) To convert a number in base two,eight  or five to a number in base ten :

• Write the number in expanded notation .
• Find the sum of the value of all the digits.

(b) To convert a number in base ten to a number in base two,eight or five,follow the steps below :

• Divide the number repeatedly by 2, 8 or 5,until the quotient is 0.
• Write the remainders from the bottom to the top.

*Converting numbers in a certain base to numbers in another base

( I ) Base two to base eight
      To convert a number in base two to base eight :
Method 1



Method 2

• Group the digits into group of three,starting from the right end.
• Substitute each group by an equivalent digit in base eight..

( II ) Base eight to base two
       To convert  a number in base eight to base two :Method 1


Method 2Substitute each digit by three corresponding digits in base two.


( III ) Base two or eight to base five
        To convert a number from base two or base eight to base five,we have to convert the number to base ten first. 



( IV ) Base five to base two or eight
        To convert  a number from base five to base two or base eight,we have to convert the number to base ten.





*Addition and subtraction of two numbers in base twoComputations involving addition and subtraction of numbers in base two can be done in the same manner as computations in base ten. 

 


 Chapter 13 : Graphs of Functions II






13.1 : Graphs of functions
* Drawing the graphs of functionsTo draw the graph of a function,follow the steps below.

• Construct the table of values for given range of the value of x.
• Choose suitable scales for the x-axis and the y-axis.
• Plot the points representing each pair of the values of x and y in the table.
• Draw a straight line or a smooth curve to join all the points. 

Characteristics of graph of functions :

( a ) Graph of linear function              

( b ) Graph of quadratic function             

( c ) Graph of cubic function
             

( d ) Graph of reciprocal function          




* Values of x and y from graphWhen given a value of x , we can find the value of y from a graph and vice versa.
   


* Shapes of graphs and graphs and types of functionsThe graph of each function has its unique shape as shown in table below.        

* Sketching graphs of functions 
 ( a ) To sketch a linear function y = ax + b :

• Determine the shape of the graph.

                

• Mark the constant b as the y-intercept.
• Substitute y = 0 to find the x-intercept.
• Mark the intercepts and draw a straight line through the intercepts.

( b) To sketch a quadratic function y = ax² + bx + c,a ≠ 0 :

• Determine the shape of the graph.

                        

• Mark the constant c as the y-intercept .
• substitute y = 0 to find the x-intercepts . Mark the intercepts.
• Sketch the graph.

( c ) To sketch a cubic function y =ax³ + b,a ≠ 0 :

• Determine the shape of the graph

                

• Substitute x = 0 to find the y-intercept.
• substitute y = 0 to find the x-intercept.
• Mark the intercepts and draw a smooth curve through the intercepts.

( d )To sketch a reciprocal function y = a , x ≠ 0 :
                                                      x

• Determine the shape of the graph.

        

• Find two points on the graph by substituting two values of x ( one of positive value and another of negative value,for instance, x = 1 and x  = -1 ).
• Mark the points and draw two separate curves through each of the points.

13.2 Solution of Equations by the Graphical Method
*Solving an equation by finding the point(s) of intersection of two graphsTo solve an equation by graphical method,follow the steps below :

• Determine the two graphs to be drawn.
• Draw the two graphs on the same axes.
• Determine the solution by reading off the values of x-coordinate of the point of intersection of the two graphs.

13.3 Region Representing Linear Inequalities in Two Variables
*Determining whether a given point  satisfies an equation or an inequalityTo determine whether a given point (x. y) satisfies an equation or an inequality, substitute the values of x and y into the equation or inequality and then compare the values of both sides of the equation or inequality.


*Position of a given point relative to the graph y = ax + b

• All the points in the region above region above the graph y = ax + b satisfy the inequality y > ax + b 
• All the points in the region below the graph  y = ax + b satisfy the inequality y < ax + b.

*Regions representing linear inequalities

• For the region representing y ≥ ax +  b or y ≤  ax + b , the line y = ax + b is drawn as a solid line.
• For the region representing y > ax + b or y  < ax + , b the line y = ax +b is drawn as a dashed line.
• The diagram below show the regions that satisfy some inequalities.

                



*Determining the region satisfying two or more simultaneous linear inequalitiesThe region satisfying two or more simultaneous linear inequalities is the common region satisfying all the inequalities.


 Chapter 15 : Matrices



15.1 : The concept of Matrices

*Forming a matrix

• A matrix is a rectangular array of rows and columns of numbers,enclosed in brackets.
• A row is a horizontal  arrangement of the numbers in a matrix.
• A column is a vertical arrangement of the numbers in a matrix.

                                

• The information given in tables can be extracted and written in the form of matrices.

*The number of rows and columns and the order of a matrix

• A matrix which has m rows and n columns is a matrix of order m x n.

              

• A matrix that contains only row of numbers is known as a row matrix.
• A matrix that contains only one column of numbers is known as a column matrix.
• A matrix that contains the same number of rows and columns is known as a square matrix.

   

*Elements in a matrix

• The numbers in a matrix are known as the elements.
• For a matrix of order m x n,each element in the matrix is identified as aij , where i is the row and j is the column.

            

• Bold capital letters such as A, B, C, ... are used to denote a matrix.The elements of the matrix is denote using the lower case from of letters such as a,b,c..

         
         The elements of the matrix are denoted by mij , m11 = 3, m12 = -1, m21 = 2, m22 = 0.




15.2 : Equal Matrices


*Determining whether two matrices are equal

Two matrices are equal if and only if
(a) the orders of two matrices are the same,
(b) all the corresponding elements are identical.


*Solving problems involving equal matrices

For two equal matrices, the unknown elements can be determined by comparing their corresponding elements.




15.3 : Addition and Subtraction of Matrices 
*Determining whether addition and subtraction can be performed on two given matrices

Addition and subtraction can only be performed on matrices of the same order.

*The sum or difference of two matrices

(a) To find the sum or difference of two matrices , make sure that both matrices are of the same order.
(b) For two matrices  A and B, 

•  A + B is obtained by adding each pair of the correponding elements.
•  A - B is obtained by subtracting each element in B from its corresponding element in A

*Addition and subtraction on a few matrices

When performing addition and subtraction on a few matrices,the corresponding elements are added or subtracted from left or right.

*Solving matrix equation involving addition and subtraction

• Finding unknown elements
• Find the values of m and n, given that
• a)                                            b)

                                                                              


            Solution :
       a)  b)                                  

• Finding unknown matrix
• Step 1 : Express the unknown matrix as the subject of the equation
• Simpllify to obtain a single matrix.

15.4 : Multiplication of a matrix by a Number    
*Multiplying a matrix by a number

(a) when a matrix is multiplied by a number , every element in the matrix is multiplied by that number.
          
(b) The multiplication of a number is known as scalar multiplication.




*Expressing a matrix as multiplication of a matrix by a number

When a matrix is given,we can express the matrix multiplication of the matrix by a number that is the common factor of all the elements in the matrix.


*Calculation on matrices

Calculation involving addition,subtraction and scalar multiplication can only be performed on matrices as follows :

• Perform the scalar multiplication
• Perform addition or subtraction from the left to the right. 

*Solving matrix equations

To find the unknown elements in a matrix equation involving addition, subtraction and scalar multiplication :

• Simplify the equation to obtain two equal matrices.
• Compare the corresponding elements in the two equal matrices.
• Solve the equations to get the values of the unknown elements.

15.5 : Multiplication of Two Matrices

*Determining whether two matrices can be multiplied and state the order of the product when the two matrices can be multiplied

Two matrices , A and B can only be multiplied if the number of columns in matrix A is the same as the same number of rows as matrix A and the same number of columns as matrix B.
               


*The product of two matrices

• When matrix A of order m × n is multiplied to matrix B of order n × p, the order of the product C is m× p.
• The element of C in the i the row and the j column, cij , is the sum of the products of the elements in the i  row of A and the corresponding elements in the j column of B.

*Solving matrix equations involving multiplication of two matrices

To solve matrix equations involving multiplication of two matrices follow the steps below :

•  Simplify the equation to two equal matrices.
• Compare the corresponding elements to find the values of the elements.

15.6 : Identity Matrices

*Determining whether a matrix is an identity matrix

• An identity matrix, I is a square matrix
• when a matrix is multiplied by an identity matrix,the product is the matrix itself.

For a matrix A, AI = IA = A,
                         I is an identity matrix for A.

*Identity matrices of any order

• There is only one identity matrix for each order matrices.
• to write an identity matrix :

(a) All the diagonal elements from the top left to the bottom right are 1.
(b) The rest of the elements are 0.

• Identity matrix of order 1 × 1 is (1)
• Identity matrix of order 2 × 2 is   
• Identity matrix of order 3 × 3 is      
• Identity matrix of order 4 × 4 is     

15.7 : Inverse Matrices

*Determining an inverse matrix

• If matrix B is the inverse of matrix A and matrix A is the inverse of matrix B then

                         AB = BA = I

• The inverse matrix of A

                            AA ^-1 = A^-1A = I

• Inverse matrices exist only for square matrices.

*Finding the inverse matrix of a 2 × 2 matrixThere are two methods to find the inverse matrix of a 2 × 2 matrix

• Solving simultaneous linear equations.
• using formula.

           


15.8 : Solving Simultaneous Linear Equations using Matrices

*Writing simultaneous linear equations in matrix form

• To write simultaneous linear equal matrices.
• Step 1 : Express  the equations as equal in matrices.
  Step 2 : Write the equations in matrix form

• Write the questions in matrix form.

                    ap + bq = h
                    cp + dq = k


*Finding the matrix  in 
                       *   When given the equation  , matrix  can be determind by using the inverse matrix of 
                


*Solving simultaneous linear equations using matricesTo solve simultaneous linear equations using matrices, follow the steps below :

• write the simultaneous linear equations i matrix form      

• Solve the matrix equation using inverse matrix              

• List the value of x and of y.

*Solving problemsMany problems in your daily life can be be written as simultaneous linear equations.We can then solve the simultaneous linear equations using matrices.

                            Chapter 16 : Variations

16.1 : Direct Variation
* Changes in a quality with respect to changes in another quantity

In every day life,changes in a quantity often lead to changes in another quantity.
We say that one quantity changes with respect to the other quantity.


* Determining direct variation

-A variable y varies directly as variable x if the ratio y   is constant.
                                                                      x                         -The relation ,y varies directly as x is written as y x x.-When variable y varies directly as x , the graph of y against x is a straight
 line passing through the origin.                 

* Direct variation in the form of an equation

-If y varies directly as x,then y is a constant.
                                        x
-The constant y is represented by k which is known as the constant of variation.
                    x
                        

-To express a direct variation in  the form of an equation :

• Write y  x  x in the form of equation y = kx, where k is constant.
• Substitute the given values of y and x into y = kx to find the value of  k.
• Substitute the value of k into y = kx.

*Value of a variable in a direct variation
-The value of variable in a direct variation can be determined when the value of the other variables is given.
-We can use two methods to find the value of a variable in direct variation,when sufficient information is given

Method 1
Using the equation form : y = kx

Method 2
Using the proportion form :   y1  =   y2
                                      x1        x2
y   x   xⁿ ( n = 2, 3,1 )
                          2 
y = kx¹¹ , k is a constant of variation
The graph of y against x¹¹ is a straight line passing through the origin , where k is the gradient.                

When sufficient information is given,the value of variable x or can be determined using :
Method 1                         Method 2
Equation form                   Proportion form
   y = kx                             y1  =   y2
                                                (  x1 )n     ( x2 ⁿ  )


16.2 : Inverse Variation
*Changes in a quantity with respect to changes in another quantity 
In everyday life,there are situations where a quantity increases as another quantity decreases.

10 workers required to complete a job in 5 days.When the number of workers increases,the time taken complete the job decreases , the time taken to complete the job increases.

              


*Determining inverse variation
-A variable y varies inversely as variable x if the product xy is a constant.
-The relation y varies inversely as x is written as y x 1 .
                                                                        x
-When variable y varies inversely as x the graph of y against 1 is a straight line passing through the origin.
                                                                                  x
                                       



*Inverse variation in the form of an equation

-If y varies inversely as x , then xy is a constant.
-The constant xy , is represented by k which is known as the constant of variation.

           


*Value of a variable in an inverse variation
-When the value of a variable in an inverse variation is given we can determine the value of the other variable
-We can use two methods to find the value of a variable in a inverse variation when sufficient information is given.

Method 1
Using the equation form : y =  k
                                            x
Method 2
Using the proportion form : x1 y1 = x2 y2




*Solving problems
-y varies inversely as x¹¹ where n = 2,3,1 if  x¹¹y is a constant.
                                                      2
          
               

^{-The graph of y against 1 is a straight line passing through the origin where k is a gradient.}
                                xⁿ
         


-When sufficient information is given, the value of variable x or  y can be determined using :

Method 1
Equation form : y = k 
                                 x<sup>n
Method 2
Proportion form : (x1) ny1   =  (x2) ny2

16.3 : Joint Variation</sup>*Representing a joint variation using symbol-The joint variation is the relationship among three or more variables in such a way that one variable varies directly and / or                 inversely as the other variables.
-The table below shows the types of joint variations.               


*Joint variation in the form of an equation 
To write a joint variation in the form of an equation :

• Determine the value of k, the constant of variation.
• Substitute the value of k into the equation. 

            

*The value of a variable in joint variation

To calculate the value of a variable in a joint variation when sufficient information is given :

Method 1 :

• Express the variation in the form of equation.
• Find the value of the required equation substituting the values of the variables into the equation.

Method 2 :

• Equate the constant, k , for two given sets of values.

               

• Find the value of the required variable by substituting the values of given variables.

    Chapter 17 : Gradient and Area under a Graph 


17.1 : Quantity Represented by the Gradient of a Graph


*Stating the quantity represented by the gradient of  a graphThe gradient of a graph represents the rate of change of the quantity on the vertical axis (y-axis) with respect to the change of the quantity on the horizontal axis 
(x-axis) 




*Drawing a distance-time graph-A distance-time graph can be drawn when given :

(b) A table of distance - time values        
(b) A relationship between distance and time  
-To draw a distance - time graph :   
(a) Use a suitable scale to mark the time on the horizontal axis and the distance on the vertical axis.
(b) Plot the points from the table of values.
(c) Joint the points. 



*The gradient of distance time-graph The gradient of a distance - time is the rate of change in distance with respect to time.
                   


*Speed for a period of timeFor a distance - time graph which consists of a few straight lines,the gradient of each straight line represents the speed for the period of time.
The distance-time graph shows the journey of a car from town X .
    

(a) Line XP - positive gradient.
                     - journey away from town X.
(b) Line PQ - gradient = 0.
                     - car is not moving.
(c) Line QR - negative gradient .
                     - return journey back to town X.

*Graph between two variables and its gradientBased on our life situations,we can draw a graph to show the relationship
between any two variables and state the meaning of its gradient.




17.2 : The Area under a Graph

*Stating the quantity represented by the area under a graph 

• The area under a graph represents the product of quantity on the vertical axis (y-axis) and the quantity on the horizontal axis (x-axis).

(a) Speed-time graph
                     
Area under the graph
= Speed (m s ^-1) × Time (s)
= Distance travelled (m)         
                           

(b) Acceleration-time graph
                     
Area under the graph
= Acceleration ( m s ^-2) × Time ( s )

= Speed ( m s ^-1 )                            

• The area under a graph may or may not represent any meaningful quantity.

(a) It is meaningful if the product of the units of both axes is the unit of measurement of a particular quantity.
(b) It is not meaningful if the product of the units of both axes is not a unit of measurement of any quantity.


*Finding the area under a graph1.The area under a graph can be determined using the formula of area of a rectangle, a triangle or a trapezium.            


 Area under the graph = Area of rectangle
                                         = b × h

    Area under the graph = Area of triangle
                                     = 1 × b × h
                                        2

   Area under the graph = Area of trapezium 
                                    = 1 × (a + b ) × h
                                       2
2. For a graph which consists of a few straight lines,divide the area under the graph into areas of rectangle, trianlge  and / 
   or trapezium and then find the sum of the area.

*Determining the distance by finding the area under a speed-time graph

           Area under a speed-time graph
            = Speed × Time
            = Distance travelled
i. Graph v = k (uniform speed) 
   The shape of area under the graph ν = k is rectangle.

ii. Graph v = kt 
   
     The shape of area under the graph ν = kt is triangle


iii. Graph v=kt+h
The shape of area under the graph v =kt + h is a trapezium



*Solving problemsWe can use the following information to solve problems involving distance,time,speed,and acceleration.

• Determine whether the graph is a distance-time graph or a speed-time graph by looking at the axes.
• The gradient of a distance-time graph is the speed.
• The gradient of a speed-time graph is the acceleration.
• The area under a speed-time graph is the distance travelled.

          Average = Total distance travelled
                            Total time taken

Chapter 18 : Probability II

18.1 : Probability of an Event

*The sample space of an experiment with equally likely out comesThe sample space, S , of an experiment with equally likely outcomes is a set of all the possible outcomes of the experiment.

*The probability of an event with equiprobable sample  space

• A sample in which each outcome is equally likely to happen is called the equiprobable sample place.
• The probability of an event A,with Equiprobable sample space , S is :

                

• If P (A) = 0 , then A is an impossible event.

• If P (A) = 1, then A is an sure event.

*Solving problems In our daily life, there are many situations where we make decisions based on th chance or likelihood of an event  happening.




18.2 : Probability of the Complement of an Event
*The complement of an event

• The complement of an event A, denoted as A' is a set containing all the outcomes of the sample space that are not the outcomes of event A.
• The complement of an event can be stated in 

           (a) words,            (b) set notation.
.

*The probability of the complement of an eventThe probability of the complement of an event A,is given  by :
                  P(A') = 1  - P (A) 

18.3 : Probability of Combined
*Listing the outcomes for combined eventsThe outcomes for combined events :
(a) 'A or B' are the elements of the set A U   B,
(b) 'A and B' are the elements of the set A ∩  B.


*The probability of combined events 'A or B' and 'A and  B'.1. The probability  for combined events 'A or B' can be determined by using the formula :
(a)
If events A and B are mutually exclusive, A ∩ B  = Ø , then
          P (A U B ) = P (A) + P (B)

( b )
 
If events A and B are not mutually exclusive, A ∩ B   ≠  Ø , then         P (A  U B ) = P (A) + P (B) - P (A ∩ B )2.
For combined events 'A and B' if the events A and B are independent,then
            P (A  U B ) = P (A) X P (B)

 
                                Chapter 1 9: Bearing




19.1 : Bearing

* Main compass directionsThe diagram below shows the eight main compass directions.

* Compass angle

• A compass direction can be stated as a bearing.
• Bearing is written as a three-digit number from 000° to 360°. For angles that are less then 100° , a zero is added on the left to form a three digit number.
• Bearing is always measured from the North in a clockwise direction.

Bearing point Q to point P is the angle between the north
direction through P and the line PQ,measured in a clockwise direction.
             

• For angles involving degrees and minutes,the angles are stated in degrees correct to one decimal point.

* Drawing a diagram to show the direction of a point relative to another point  To draw a diagram showing the direction of point Q relative to point P.Follow the steps : 

• Mark point P and draw a vertical line through P to represent the north direction,N.

• By using a protractor,measure NPQ in a clockwise direction from the north line.
• Mark and label Q.
• Draw a line to join points P and Q

* Stating the bearing of a point from another point When the bearing of point Q from point P is given we can
 find bearing of point P from point Q using these two methods.


* Solving Problems

• The knowledge about bearing can be used to solve problems involving navigation and map raeding.
• To solve problems involving bearings :

             (a) Draw a diagram to represent the given situations 
             (b) Solve the problem using the Pythagoras' thoerem and/or trigonometric ratios. 

    Chapter 20 : Earth as a Sphere





20.1 : Longitude

*Great circle

A great circle is a circle on the surface of the earth with centre O, where O is the centre of the earth.

(a) All circle through th North pole, N, and the South pole, S are great circles.
(b) The equator is also a great circle              


*Longitude of a given point

• The half of a great circle that joins the North pole and the South pole is known as a meridian.
• There are two meridians on a great circle through both poles.

               

• The meridian through Greenwich in England is known as the Greenwich Meridian with longitude 0° 
• The longitude of the other meridians are determined by the angle between the plane through the meridian and the plane of the Greenwich Meridian,measured either east or west of the Greenwich Meridian.
• In the diagram below,NGS is the Greenwich Meridian.

 

*Sketching and labelling a meridian with the given longitude

Toe sketch and label a meridian with longitude 50°  E,follow the steps below.

• Draw a cirle with centre O to represent the earth.
• Draw a straight dashed line,NOS to represent the earth's axis.

              =

*Difference between two longitudes

The difference in longitude between two meridians can be determined by :

(a) Adding the angles of the two longitudes,if the meridians are on opposite sides of the Greenwich Meridian              
Difference between the two longitudes
= QOR
= 25° + 78°
= 103°


(b) Subtracting the angles of the two longitudes if both the meridians are on the same side of the Greenwich Meridian.                    
Difference between the two longitudes
=  BOC
= 92°- 48°
= 44°



20.2 : Latitude

Latitude shows whether a place is located to the north or to south of the equator.

*Circle parallel to the equator

(a)The great circle perpendicular to the polar axis is known as the equator.
(b)The other circles which are parallel to the equator are known as the parallel of latitude.                        

(c)To sketch a circle parallel to the equator ,follow the steps below :

• Draw a circle with centre O to represent the earth.
• Draw two straight dashed lines, NOS to represent the earth's axis and AOB to represent the diameter of the equator.

                         

• Draw a straight dashed line PQ parallel to AOB to represent the diameter of the parallel of latitude.

                      

• Sketch a circle with AB as its diameter for the equator and circle with PQ as the diameter for the parallel of latitude.

                       


*Latitude of a given point

• The latitude of the equator is 0°
• The latitudes of the other parallels of latitude are measured either north or south of the equator.
• The latitude of a parallel of latitude is determined by angle at the centre of the earth is subtended by the arc of a meridian from the equator to the parallel of latitude. 
• The latitude of a parallel of latitude x° to the equator of the equator is x°N
• The latitude of a parallel of latitude y° to the south of the equator is y°S.

           


*Sketching and labelling a parallel of latitude

To sketch and label a parallel of latitude with latitude 35°N,follow the step below :

• Draw a circle straight dashed lines NOS to represent the earth's axis and AOB to represent the diameter of the equator.

       

• Mark and label AOP = 35° to the north of the equator and draw a straight dashed line PQ parallel to AOB.

  

• Sketch the equator and the parallel of latitude that passes through point P 

*Difference between two latitudes

The difference in latitude between two parallels of latitude can be determined by :

(a) Adding the angles of the two latitudes if the parallels of latitude are on opposite sides of the equator or on different                                hemispheres.

              


(b) Subtracting the angles of the latitudes,if both the parallels of latitude are on the same side of the equator or on the same                      hemisphere.            



20.3 : Location of a Place

*Latitude and longitude of a place

• A place on the surface of the earth is represented by a point
• The location of a place on the surface of the earth is written as an ordered pair of latitude and longitude ( latitude,longitude ).
• The location of a place A at latitude x°N and longitude y° E is written as ( x°N, y°E ).

        


*Marking the location of a place

A place on the surface of the earth is marked as the intersection point of the latitude and the longitude of the point.
The location of  a place on the surface of the  earth is written as an ordered pair of  latitude and longitude ( latitude , longitude ).
The location of a place A at latitude x°N and longitude y°E is written as ( x°N , y°E )

*Marking the location of a place

A place on the surface of the earth is marked  as the intersection point
of the latitude and the longitude of the point.


*Sketching and labelling the latitude and longitude of a place

To sketch and label the latitude and longitude of place P ( 40°N,53°W ) follow the steps below : 

Draw a circle with centre O to represent the earth .Sketch and label the Greenwich Meridian,NGS and the equator. 




Mark an angle of 53° to the west of the Greenwich Meridian.Sketch and label meridian 53°W. 



Mark angle 40°to the north of the equator.Sketch and label the parallel of latitude 40°N.



Mark and label point P at the intersection of the meridian 53°W and the parallel 40°N

         


20.4 : Distance on the surface of the earth

*Length of an arc of a great circle in nautical mile

• The distance between two points on the surface of the earth is the length of the arc that joins the two points.
• The distance between two points on the surface of the earth is measured in nautical miles.
• One nautical mile (n.m) is the length of an arc of a great circle which subtends an angle of 1'(1 minute) at the centre of the earth.
• Based on the diagram below,the meridian NABS and the equator are great circles.

             


* Distance between two points along a meridian

For points A and B along a meridian,
         
  
where Ø is the angle subtended by arc AB at the centre of the earth.



*Latitude of a pointThe latitude of a point can be determined given the latitude of another point and the distance between the two points along their common meridian.

*Distance between two points along the equator

We can find the distance between two points measured along the equator if the longitude of both points are given,For two points A and B the equator,
                                          

*Longitude of a point along the equator

The longitude of a point can be determined given the longitude of another point and the distance between both points along the equator.


*Relation between the radius of the earth and the radius of the parallel of latitude

In the diagram below,P is the centre of the parallel of latitude and O is the centre of the earth.Let r be the radius of the parallel of latitude Ø°N  and R  be the radius of the earth.




*Relationship between the length of an arc on the equator and the length of the corresponding arc on a parallel of latitudeIn the diagram below, A and B are two points on the parallel of latitude,Ø°N, and P and Q are two points on the equator.O' is the centre  of the parallel  of latitude  Ø°N  O is the centre of the earth.


*Distance between two points along a parallel of latitude Ø°N (or Ø°S),                 

*Longitude of a point along a parallel of latitude

The longitude of a point can be determined given the longitude of another point and the distance between both points along a parallel of latitude.

*Shortest distance between two points on the surface of the earth 

The shortest distance between two points on the surface of the each is the distance measured along a great circle.

*Solving problems 
We can use the knowledge involving the distance between two points on the surface of the earth to solve problems involving distance and travelling between two places on the surface of the earth.In the field of aviation and navigation,the unit for speed is knot.
     
               1 knot = 1 nautical mile per hour

     Chapter 21 :Plans and Elevations



21.1 : Orthogonal Projections

*Identifying orthogonal projections

The orthogonal projection of an object on a plane is the image formed
by the normals from the object onto the plane.          


*Drawing orthogonal projections

To draw the orthogonal projection of a given object on a plane :
(a) Draw the normals from each verticles of the object to the given plane.
(b) Join all the points on the plane.


*Determining the difference between an object and its orthogonal projection

1. The length of the edges and the size of the angles between an object and its or the angles between an object and its orthogonal        projection are the same,if the edges are parallel to the plane of the orthogonal projection.
     The diagram below shows the orthogonal projection of the given prism on a vertical plane as viewed from X.

   The length of the edges AB , BF , FE and AE of the object are equal to that of the orthogonal 
    projection as these edges are parallel to the vertical plane.
 

2. The length of the edges and the size of the angles between as object and its orthogonal projection are different,if the edges are not parallel to the plane of the orthogonal projection.

The edges FB and GC of the object are different from that of the orthogonal projection , as these edges are not parallel to the plane of the orthogonal projection.
ABF and BFE of the object are also different from that of the orthogonal projection. 

3.Determine whether the length of PQ in each of the following objects and its orthogonal projection on the horizontal 
   or vertical plane as viewed from X are the same.


21.2 : Plans and Elevations


*Drawing the plan of solid objects

The plan of an object is its orthogonal projection on a horizontal plane (as viewed from above ).           


*Drawing the elevations of solid objects 


1. The elevation of an object is its orthogonal projection on a vertical plane.
2. The elevation as viewed from the front of the object is known as the front elevation
3. The elevation as viewed from the side of the object is known as the side elevation.
4. The plans and elevations can be drawn using alternative methods as given below :
   (a) Divide a piece of plain paper into 4 quadrants.
   (b) Draw an angle bisector in the third quadrant.               
  (c) Draw the plan of the solid in the fourth quadrant
  (d) Draw the front elevation ( as viewed from V ) in the first quadrant,using projection lines from the plan.
  (e) Draw the side elevation (as viewed from W) in the second quadrant, using projection lines from the plan and the front
         elevation.

5.  If the solid is viewed from left to right then the side elevation is drawn to the right of the front elevation.           


*Drawing the plan and elevations of a solid object to scale

In the scale of 1 : n,
               

We  can draw the plans and elevations of large solid objects such as buildings using a suitable scale.