Angles of triangles and polygons

Hello students here we are going to discuss the angles of triangles and angles of polygons. But before discussing it you should get familiar with the term 'angle'. If two rays OP and OQ have a common end point ‘O’ then it form angle POQ, written as ∠POQ.

Angle may be interior, if any point in its (∠ POQ) plane lies on the same side of OP as Q and also on the same side of OQ as P and exterior of an angle has points in its plane which do not lie on the angle or in its interior.

Angle is of following types such as: right, acute, obtuse, straight, reflex and complete angles.

Let us take a look at Angles of triangles: Suppose we have a triangle ∠ABC, the interior of this triangle is the set of all points in its plane, which lies on the same side of OP as ‘Q’ and also on the same side of OQ as P or in other words if any angle is inside the boundary of the triangle then it said to be interior angle of the triangle and if any angle is outside the boundary of the triangle then it said to be exterior angle

Note: - The interior angles of a triangle can add up to the 180°. (know more about icse class 10 books, here)

Angles of polygons: Polygons are the shapes that have more than two angles or have straight sides. Example triangle, rectangle, and rhombus, etc. Interior angles are those which lie inside the boundary and vice versa.

Note: The exterior angles of a polygon can add up to the 360°.

 Above information will be useful for the Grade IX students.

In the next session we are going to discuss Error and magnitude.

.

Units of measurement

Units are defined for physical quantities and units of measurements are nothing but definite magnitude of these physical quantities. These units of measurement are derived and adopted after some conventions or laws and now we are using these units as a standard for measurement of the physical quantity.
There are various systems for defining units of measurement:

1.       Traditional System: It was used in ancient time.

2.       Metric Systems: It is the system which is being used universally. Many metric systems have been evolved after the adoption of original metric system in 1791. But the standard metric system which is globally used is the International Systems Of Units (SI), it is also known as the modern form of metric system.

3.       Natural Systems: This system consists of unit values which occur naturally in science. They include atomic units (au), solar mass, the megatons etc.

Here are seven units of measurement which are evolved or derived as a basic set from which all other SI units are derived and their corresponding physical quantities:

1.       Length                            meter (which represents a definite predefined length , i.e., if we say 5                                                         Metre or 5m, we mean 5 times the predefined length called “meter”)

2.       Time                                   seconds

3.       Electric Current                 Ampere

4.       Mass                                    Kilogram

5.       Temperature                      Kelvin

6.       Luminous Intensity           candela (cd)

7.       Amount of substance           mole (mol)

We can derive any other physical quantity by simply multiplying units of measurement.

Units of measurement have been playing a very significant role in various tasks since 1791: like fashioning clothing, construction of dwellings of definite size and shape, physics, medicine, metrology etc.

 Here is one example of how the derivation of other units takes place from the above seven units of measurement:
Area – the SI unit is meter square (m²)
A=Length * breadth (area of rectangle)
A= m * m = m² (meter square)
As SI unit of length and breadth is meter (m).
Like that we derive many more units.

In the next session we are going to discuss Error and magnitude.

different properties to solve problems in Grade IX

In the mathematical field there are a number of properties available to solve the problem related to math subject. We can use different properties to solve problems in math. In the general sense, the properties to solve problems include distributive property, identity of addition, identity of multiplication, inverse of multiplication, associative property, commutative property etc. Here we are going to discuss about the different properties to solve problems.
There are various types of properties, which are elucidated with their explanation in this session. This part helps the students of Grade IX. These properties help to solve maths problems in an efficient manner.
1) Commutative property:  a + b + c = c + b + a
2) Distributive property:   a(b + c) = ab + ac
3) Associative property:  (a + b) + c = a + (b +c)
Let’s show you how to use these properties with solving quadratic equations.
Quadratic equations are those equations which are polynomial equations of the second degree. Here we show you the form of quadratic equation:
   ax² + bx + c = 0
Here x variable is unknown variable, with the constants as a, b and c with a not equal to 0.
Now we show you how to solve the inequalities of quadratic equation. In these we use the different properties of maths to solve problems.  ( to know more about central board of secondary education, here)

Example: Solve for x in given expression 2x² + 4x = x² – x – 6?
Solution: y = 2x² + 4x
               Y = x² – x – 6
       In that part we perform the operation of simplification by properties to solve problems. Let’s show you below:
              2x² + 4x = x² – x – 6
              2x² + 4x- x² + x + 6=0
              x² +5x + 6 =0
Now we perform the factorization of equation to solve it.
                x² +5x + 6 = 0
              (x + 2)(x+3) =0
So we can say that:
          x = -2 and x = -3
In the next session we will discuss about Units of measurement. 

Arc Lengths

Arc lengths:
Let us talk about arc length for grade IX students. An arc length is the length of the part of the circumference of any circle. For measuring arc length, we will first understand the concept of radian.
The equation of radian is:
l=rθ
here, l represents the length of the arc.
Every arc will make an angle θ at the center of the circle whether it is large or very small.
The ratio of the circumference to the diameter of the same circle represents the radian.
In mathematical form,
∏=circumference/diameter      ------------- (1)
Now, as we know that
diameter=2*radius
putting In equation 1, we get
∏=circumference/2*radius.
Or, 2∏=circumference/radius
It is one revolution of a circle. One revolution of a circle can be divided into four right angles. The circumference covers these four right angles. (know more about cbse board, here)
Here,
∏ stands for radian measure for 1 revolution.
The arc length formula is:
θ =l/r,
where, l=arc length
r=radius of circle.
radian measure(θ)=l/r
The ratio l/r is a real number and it is calculated as an angle (θ) at the center. The angle (θ) is measured in radians.
Let us discuss a theorem below,
Theorem: if the ratio of the arc length to the radius of circle is same for the two circles, then they must possess same angle θ at the center.
In mathematical terms,
if l1/r1=l2/r2
then,
θ1=θ2
Let us take an example to calculate arc length.
Example 1: calculate the arc length of a circle if it is given that the radius is 10 cm and central angle is 2.35 radians.
Solution:
As we have discussed earlier,
θ=l/r
l is the arc length of the circle
hence,
l=θ r
θ=2.35 radians,
θ is already in radians.
If it is given in degrees, we will have to first change it into radians.
L=2.35*10=23.5 cm.
Let us discuss another example based on the theorem above.
Example 2: Two circles have the same angle θ at the center created by their arcs.
It is given that r1=20 cm
r2=15 cm
l1=25 cm
l2=?
Solution:
we have to calculate the length of the arc of the second circle.
As given θ1=θ2=θ,
using the theorem,
l1/r1=l2/r2
it implies,
l2=(l1/r1)*r2
l2=(25/20)*15
l2=18.75 cm
In the next session we will discuss about different properties to solve problems in Grade IX

Direct indirect measures

Direct and indirect measures in maths for Grade IX:
Direct measure is a measurement of something that we actually want while the indirect measurement is that we are measuring something other than the actual thing of interest.
Let us learn Direct and indirect measures through examples:
Direct measure is looking at actual samples of students’ work produced in the programs organized by the school authority while indirect measurement is gathering information from the sources other than looking at the actual programs those were attained by the students. (know more about cbse sample papers, here)
A direct measurement is counting the number of people those who have entered in a stadium through the gate while the indirect measurement is counting the number of people entering in the stadium with the help of tickets sold from the ticket window. It can be false or not accurate because of all the tickets those that were sold, it is not necessary that they have been used to get entry in the stadium.
If we count the number of students using term papers taken by them by the number of copies then it is the direct measurement. If the number of students is counted on the paper basis then it is indirect measurement because all students may not have taken the exam.
Written surveys and questionnaires are also the examples of indirect measurement.
Let’s have another example. If a person goes to a doctor than he might get a form to fill in which some questions are like:
• Any person in your family affected by cancer.
• Is that person taking medicines like painkiller regularly?
• Is that person is allergenic to anything.
This is like the survey and definitely would not give accurate result. It’s just a measurement or survey that will provide the general information related to the population. A lot people fill these types of forms. So a general percentage can be calculated but it might not be accurate. Accuracy is far away from the result drawn.
In the next session we will discuss about Arc Lengths.

Rate, distance in geometry

In the mathematics we study a lot of concepts which are related to our daily life. These concepts help to solve the problems related to mathematical calculations. In the mathematical field there are many ways to represent the things in the measurable quantity. These measurable quantities provide an easy and convenient way to represent the things, which help in to performing the various tasks related to our daily routine life.
Here we are going to discuss about the rate and distance in geometry. Rate contains the collection of several measurable quantity or measures to specify the quantity of the product or things. In simple terms rates is a ratio, which demonstrates the relationship of a thing to the measurable quantity.

Suppose John makes a trip to Hollywood, in which he covers a distance of 250 miles in 5 hours then what is the speed of his car?
This would be represented by dividing the distance by the time.
            Let’s see                    250/5 = 50 miles per hour
It means john covers the distance at the rate of 50miles/hour. Here ( / )symbol represent ‘ at per’.
Now we show the relationship between rate and distance. As we describe rate termed as ratio, which represents the comparison of two different numbers to display them in measurable quantity.
Now we will see an example to represent the relationship between the rate and distance:
Example: Suppose there is a shop where jack purchases 5 liter milk. Now he pays $15 for the milk. So find the amount of 1 liter milk.
Solution: Here given that purchase milk = 5 liter
                               Paid amount             = $ 15
                         So, amount of 1 liter   =       $15 / 5
                                                                =   $3/liter
                                                 
In the next session we are going to discuss time and angle measures

time and angle measures

Time and angle measures are one of the important types of mathematical problems. We all know that time problems (clock angle measure problem) are all about finding the angles between the hands (Hands are minute hand or hour hand or second hand) of an analog clock. This is very important part of geometry. In geometry the measurement unit for angle is the “Degree”. A circle is divided into 360 equal degrees.  Moreover, degrees are divided into two terms, known as Minutes and Seconds. This division of degree is not universal. (To get help on central board of secondary education books click here)
These problems generally relate to two different measures which are angles and time. For such problems we always consider the change rate of the angle in degree per minute (deg/min) and we may also use arc lengths in this. In an analog clock the hour hand of the clock turns 360 degree in 12 hours which are 720 minutes or we can say 0.5 degree per minute. In the same analogue clock the minute hand turns 6 degree per minute or 360 deg in 1 hour (60 minutes).
Now to solve related problems we need to apply some formulas in form of general equations. These equations are:
1.      Equation for angle of the hour hand of the clock:
Ahr = ½ Me = ½ ( 60 H + M)
Here Ahr is the angle of the hand measured clockwise from the exact time 12’o clock. M is the minute past hour and Me is minute past 12’o clock.
2.      Equation for degree on minute hand of the clock:
Amin  = 6*M
Where Amin is the angle of the hand measured clockwise from the time 12’o clock and its unit is degree.
For example: Time given is 5:30 then the angle of hour hand in degrees:
Ahr  = ½(60*5 + 30) =165 degree
Similarly for the same time the angle of minute hand in degree:
Amin  = 6*30  = 180 degree.
So today we learnt about Times and Angle measures. In the next session I will tell you about arc lengths and In the next session we will discuss about Rate, distance in geometry.