Basic constructions

In the previous post we have discussed about Triangle inequality theorem and In today's session we are going to discuss about Basic constructions. Geometry is an important part of the mathematics; in geometry we make many types of figures, angles and lines. So the construction is an art by which we can draw the shapes. Basic constructions in geometry include the construction of the circles, triangles, lines and many types of shapes. And for the basic construction we use some steps that are to be followed for making shapes and also we need some instruments such as compass, ruler, pencil, drafter, dividers, etc.
When we study about the basic constructions in geometry then first we should be familiar with two things,
First one is line bisector and second is angle bisector. With help of angle bisector method we can create many angles like 15Ëš, 30Ëš, 45Ëš, 60Ëš etc. we can construct shapes for the triangle, rectangles etc. but the line bisector and the angle bisector are basic geometry constructions so let us take a look on some steps of the line bisector.
To draw the bisector of a line let us take an example:
i) Draw a line segment AB = 6.5 cm. (know more about cbse syllabus , here)
ii) With ‘A’ as a center and radius equals to more than half of AB, draw two arcs, one above AB and the other below AB.
iii) With ‘B’ as a center and the same radius, draw two arcs, cutting the previously drawn arcs at points ‘C’ and ‘D’ respectively.
iv) Join CD, intersecting at a point AB at a point ‘P’, then, CD bisects AB at a point ‘P’.
For basic constructions of shapes we have to follow some steps.
 Grade IX students can learn basic geometry from above discussion.

Triangle inequality theorem

As we all are very well aware that geometry provides lots of shapes and figures to solve out the problems that are related to mathematics. In mathematics, triangle is a shape that provides the various concepts to solve the problem. Triangle inequality is one of them, which states that the sum of lengths of any two sides of the triangle must be greater than or equal to the length of third side. Euclidean geometry says that triangle inequality is a theorem about the distance. In simple definition of the triangle inequality theorem we can say that the total of any two sides of triangle must be greater than the third side of the triangle. If in case the above theorem is not applied at the time of triangle creation then that figure is not considered as a Triangle. This theorem helps the student of Grade IX to understand the concept.
Suppose a triangle has three sides which named as A, B and C. Now we check that the given triangle is a complete triangle or not by applying the inequality theorem. The Triangle Inequality theorem says that:
B + C > A,
A + B > C,
C + A > B,
If in the given triangle, available values satisfy the above combination or pairs of inequality then the triangle can be considered as complete triangle. In study of Triangle of geometry, we generally study three types of triangle which are equilateral triangle, isosceles triangle and right angle. In all types of triangle we can easily see the operation of triangle inequality theorem. (know more about icse syllabus 2013 , here)
Example: Can a triangle have the side lengths of side 1 = 4cm, side 2 = 8 cm and side 3 = 2cm?
Solution: Now we need to apply the triangle inequality theorem. By applying the Triangle inequality theorem we create three combinations that are given below:
Side 1 + side 3 > side 2,
  4 + 2 > 8,
The first equation get false then we can say that the above given sides are not able to form the triangle.
In the next session we will discuss about Basic constructions. 

Error and magnitude

Error and magnitude can be defined as the difference between the approximate and absolute value that means there are many situations in which the measurement of the value is not absolute due to some instrumental problems, so obtained value is approximate value, in place of absolute value for measuring the difference there are some types of methods, they give the difference between absolute and approximate value that is known as error. It helps students of grade IX to understand the difference between magnitudes of exact and approximate value that is defined as error.
There are basically two types of errors that are relative error and other is absolute error.
Absolute error: This type of error is defined as the difference between the magnitude of exact and approximate value .This is denoted as e (absolute error) = | m – m _approx |.
Where ‘m’ and ‘m _approx’ are the magnitude of exact and approximate values respectively and ‘e’ denotes the absolute error.
Relative error: When absolute error is divided by the magnitude of the exact value or absolute value is known as relative error, it will be denoted as e _relative = | m – m _approx | / m,
Or e _relative = e _absolute / m.
Percentage error: Percentage error is defined as the relative error, it is multiplied by 100 or the percentage error is a relative error that is described in form of per 100.
It is denoted as e % = e _relative * 100 = (| m – m _approx | / m) * 100.
Let us take an example,
Exact value is 90 and approximate value is 89.6 then,
Absolute error = 90 – 89.4 = 0.6,
Relative error = 0.6 / 90,
% error = (0.6 / 90) * 100,
In the next session we are going to discuss Basic constructions.

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.

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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