Learn Points and Lines

To learn about points and lines, A point used to represent a place in a plane with a help of pencil, a point is nothing but the dot , it has no dimension or no width, it’s only a simple black dot. In geometry co ordinates of a point which shows the particular place in a segment for representation.Line has two end points is called segment. Line segment is denoted with a connected piece of line.line segments names  has two endpoints and it is named by its endpoints.


learn about points and lines:

To learn about the geometric points and lines we have to know the classification of a points and lines.points and lines classification are as follows.
Collinear points:
When three or more points lies on the same line is said to be collinear points.

Midpoint:
A halfway point where line segment divides into two equal parts are called midpoint.

Equidistant point:
A point which is said to be equidistant in a line segment where point is equal length from other points which are in congruent then the point is equidistant point.

Parallel line segment:
Two lines which does not touch each other are called parallel lines.

Perpendicular line segment:
Two line segment  that form a L shape are called perpendicular lines.


learn problems in points and lines:

Example 1:
Find the distance between the points A(5,2) and B (7,3).


Solution:
Let assume "d" be the distance between A and B.           (x1,y1)= (5,2), (x2,y2)= (7,3).

Then d (A, B) =`sqrt((x2-x1)^2+(y2-y1)^2)`

= `sqrt((7-5)^2 +(3-2))^2)`

= `sqrt(2^2+1^2)`

= `sqrt(4+1)`

=`sqrt5`

Example 2:
Find co-ordinate of the mid point of the line segment joining given points A(-1,1) and B(3,4)

Solution:
The required mid point is
Formul a   `((x_1+x_2)/2 ,(y_1+y_2)/2)` here,  (x1, y1) = (-1,1),(x2, y2) = (3,4)

=  `((-1+3)/(2))``((1 +4)/(2)) `

= `(2/2) ` ,  ` (5/2)`

=`(1,5/2)`

Example 3:
Find the slope of the lines given (2,-1) and (1,3)

Solution:
(x1,y1)= (2,-1), (x2,y2)= (1,3).
We know to find slope of line,m=` (y_2-y_1) /(x_2-x_1)`

=`(3+1)/(1-2)`

m =`4/-1` = -4
Example 4:

Find the equation of the line having slope `1/2` and y-intercept −3.
Solution:
Applying the equation of the line is y = mx + c
Given,       m = `1/2` ,c = −3
y = `1/2` x + (−3)

or  2y = x − 6
or  x− 2y − 6 = 0.

Forms of Quadrilaterals

Forms of Quadrilateral

What is Quadrilateral?

A quadrilateral is a 2-dimensional closed shape with four straight sides.  E.g. The shape ABCD shown here is a quadrilateral.

A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral.  AC is a diagonal of quadrilateral ABCD, as is BD.

Properties of a quadrilateral:

Four sides (or edges)
Four vertices (or corners).

The interior angles add up to 360 degrees

Forms of Quadrilateral:

Parallelograms

Quadrilaterals are called parallelograms if both pairs of opposite sides are equal and parallel to each other.  Different parallelograms and their properties are described below.

Opposite sides of a parallelogram are parallel and equal in length.
Opposite angles are equal in size.

Rectangle

Opposite sides of a rectangle are parallel and equal in length.
All angles are equal to 90°



Square

A square has equal sides and every angle is a right angle (90°)
Also opposite sides are parallel.
A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

Rhombus

All sides of a rhombus are equal in length
Opposite sides are parallel.
Opposite angles of a rhombus are equal.

The diagonals of a rhombus bisect each other at right angles

Trapezium

A trapezium has one pair of opposite sides parallel

A regular trapezium has non-parallel sides equal and its base angles are equal, as shown in the diagram

Kite

Two pairs of adjacent sides of a kite are equal in length
One pair of opposite angles (the ones that are between the sides of unequal length) are equal in size.
One diagonal bisects the other.
Diagonals intersect at right angles.

Pyramid of Biomass

Pyramid of Biomass is the graphic representation of biomass of the successive trophic levels of an Eco system. Biomass refers to the total weight of dry matter present in the ecosystem at anyone time. The pyramid of biomass indicates the gradual reduction in biomass at each trophic level from base to top. The total biomass of producers is much more than the total biomass of herbivores. Like wise the total biomass of the third level will be less than the herbivores and so on. The pyramid of biomass are generally upright, but in case of parasites, the pyramid of biomass is inverted. Similarly, if organisms of lower levels are much smaller than those of higher levels, biomass pyramid may be inverted.                                                            

Examples of Pyramid of Biomass.1

Pyramid of Biomass in a Marine ecosystem

In the marine Eco system the Phytoplankton occupy the basic trophic level and the next trophic level is occupied by the Zoo plankton, the third and fourth are occupied by carnivores such as worms, and molluscs and carnivores fishes respectively. Here too the biomass falls in a graded manner from the basic trophic level to higher trophic level.

Examples of Pyramid of Biomass.2

Pyramid of Biomass in  a  grass land 

In a grass land the biomass of grasses is the maximum and it gradually decreases towards the consumer level. That is  the biomass is decrease in this order. Grasses have higher biomass than that of snake  and snake have higher biomass than that of Hawk.

                                                 Grass ------------> Snake -------------> Hawk

How to Rewrite Radicals

Radicals are nothing but a root which we also called square root. Square root is indicated with the symbol (sqrt) and sqrt (). Normally, radicals are rewrite like, sqrt of (a) is rewrite as a ^(1/2). Likewise radicals are expressed in various forms. The expression sqrt (8) is read as “radical eight”, or “the square root of eight”. Thus, we are going to see how to rewrite radicals in different ways.

Formula for how to rewrite radicals:

Formula for how to rewrite radicals:
General expression with exponent and radical:
                                          `( ^nsqrt (a) ) ^m`   = `( ^nsqrt (a) ) ^m`   = `(a1/n) ^m ` = `am/n`                         
Multiplication property for radical expression:  (  `^nsqrt (ab)`   ) = ( ` ^nsqrt (a)` )  ( ` ^nsqrt (b)` )
Division property for radical expression:  ( `^nsqrt (a/b)` ) = (  `^nsqrt (a)` ) / (  `^nsqrt (b)`   )
Different forms of radicals is,
                    A square (second) root is written as (  ` sqrt(x)` )

     A cube (third) root is written as (  `^3sqrt(x)`  ),

     A fourth root is written as (  `^4sqrt(x)`   ),

     A fifth root is written as:  (  ` ^5sqrt(x)` ).

Example for how to rewrite radicals:

Example for how to rewrite radicals: Rewrite radical(`sqrt (1225)` )
Given: (`sqrt (1225)` )
Solution: Given question says, radical (1225),
      When, we take radical for 1225, we obtain 25*49.
Because,                 (  `^nsqrt (ab)`   ) = ( ` ^nsqrt (a)` )  ( ` ^nsqrt (b)` )
                                       `sqrt (1225)` = `sqrt (25)` `sqrt (49)`
                                                        = `sqrt(5)`  * `sqrt(7)` 
                                       `sqrt (1225)` = `35`
Thus, we can do how to rewrite radicals in the prefered way.

Example for how to rewrite radicals: Rewrite radical( `^3sqrt (512)` )
Given: (  `^3sqrt (512)` )
Solution: Given question says, cubic root of (512),
      When, we take cubic root for 512, we obtain 8.
Because,                           `8*8*8 = 512`
                                          `^3sqrt(512)`` =` `8^3`
  Therefore, cubic root for (512) = 8^3
Thus,we can do how to rewrite radicals in the prefered way

Example for how to rewrite radicals: Rewrite radical `^3sqrt (729)`
Given: (`^3sqrt (729)` )
Solution: Given question says, cubic root of (729),
      When, we take cubic root for 729, we obtain 9.
   Because,                         `9*9*9 = 729`
                                         `^3sqrt(729) = 9^3`
     Therefore, cubic root of (729) = 9^3


Example for how to rewrite radicals: Rewrite radical `^4sqrt (1296)`
Given: (`^4sqrt (1296)` )
Solution: Given question says,  fourth root of (1296),
      When, we take fourth radical for 1296, we obtain 6.
Because,                             `6*6*6*6 = 1296`
                                             `^4sqrt(1296 )= 6^4`
     Therefore,   fourth root of (1296) = 6^4

Example for how to rewrite radicals: Rewrite radical `^5sqrt (3125)`
 Given: (`^5sqrt (3125)` )
Solution: Given question says, fifth root of (3125),
      When, we take Fifth radical for 3125, we obtain 8.
Because,                       `5*5*5*5*5 = 3125`
                                             `^5sqrt(3125) = 5^5`
     Therefore,    fifth root of (3125) = 5

           

Math Scientific Expressions:

In this article we shall discuss about math scientific expressions. Scientific notation expressions are meant by transform very big numbers into simplest form of a number. That is called as scientific notation expressions to integer. Scientific notation expressions is nothing but exponential of the base integer 10 have expressed in the number scientific notation expressions. Example for scientific notation expressions is 785000000 this can be note down as 7.85 x 10⁸. Here, 8 is the power of 10.   

Example problems based on math scientific expressions:

The example problems based on math scientific expressions to integer is given below that,

• Example 1:

How do you write 67000 in math scientific expressions?
Solution:
Step 1:

The given number is 67000.

Step 2:

The given number is not a big number, but it is simple to convert scientific notation.

First rounding 6.7 x 10⁴ = 67000 [10⁴ = 10000]  

Step 3:

The concluding answer for math scientific expressions is 6.7 x 10⁴.

• Example 2:  

How do you write 2800000 in math scientific expressions?
Solution:
Step 1:

The given number is 9100000.

Step 2:

The given number is a big number, but it is simple to convert scientific notation.

First rounding 9.1 this is not equivalent to the given number.

Step 3:

Here, 6 digits subsequent the decimal point.

So, 9.1 × (1000000)

Therefore the rounding will be 10⁶.

Step 4:

The concluding answer for math scientific expressions is 9.1 x 10⁶

• Example 3:

How do you write 843000 in math scientific expressions?
Solution:
Step 1:

The given number is 843000.

Step 2:

The given number is a big number, but it is simple to convert scientific notation.

First rounding 8.43 this is not equivalent to the given number.

Step 3:

Here, 5 digits following the number are there so we can write 10⁵.

Step 4:

The concluding answer for math scientific expressions is 8.43 x 10⁵.

.

Practice problems based on math scientific expressions:

The practice problems based on math scientific expressions is given below that,

• Problem 1:

How do you write 9510000 in math scientific expressions?
Answer: The concluding answer for math scientific expressions is 9.51 x 10⁶.

• Problem 2:

How do you write 852000 in math scientific expressions?
Answer: The concluding answer for math scientific expressions is 8.52 x 10⁵.

• Problem 3:

How do you write 75000 in math scientific expressions?
Answer: The concluding answer for math scientific expressions is 7.5 x 10⁴.

Just Math Answer

Some of the main branches of mathematics are algebra, geometry and calculus. Various number of problems exists in these branches.
In this article of just math answer, several problems related to various branches of mathematics are given with their respective answer. In addition, several practice problems with answer key are given for better understanding.


Math Example problems with answers:

Example 1:

Find the Mean of the given data set: { 2, 6, 19, 29 }

Solution:

Mean   =  ( 2 + 6 + 19 + 29) / 4

=  56 / 4

= 14

Answer: Mean  = 14

Example 2:

Find the volume of cylinder given the radius is 13 cm and 32 cm.

Solution:

Volume of cylinder = `pi` r2 h cubic units.

= (3.14) * 132 * 32

=  3.14 * 169 * 32

= 16981.12 cm3

Answer: Volume of cylinder = 16981.12 cm3

Example 3:

Find the area of the square with the side length of 9 cm.

Solution:

Area of square  =  ( a ) 2

= ( 9 )2

= 9 * 9

= 81 cm2

Answer: Area of square =  81 cm2

Example 4:

Find the area of a triangle through base of 15 m and a height of 8 m.

Solution:

Area of a triangle = ½ b h

= ½ ( 15 ) ( 8 )

= 0.5 * 15 * 8

= 60 m2

Answer: Area of triangle = 60 m2

Example 5:

Solve :  6 ( x + 3 ) = 72

Solution:

6( x + 6)  =  72

6x + ( 6 x 6 )  =  72

6x + 36  =  72

6x  =  36

x  =  6

Answer:  x =  6

Example 6:

Find lateral surface area of cone with radius 5 cm and slant height 16 cm.

Solution:

Given:  Radius r = 5 cm

Slant height l =16 cm

Lateral surface area  = `pi` r l   square units.

= 3.14 x 5 x 16

= 251.2 cm2

Answer: Lateral surface area = 251.2 cm2



Example 7:

Find the range of the data set { 25, 42, 36, 21, 48 }

Solution:

Here, The maximum value is 48

The minimum value is 21

Range  =  Maximum value – Minimum value

=  48 – 21

= 27

Answer: Range  = 27


Math Practice problems with answer:

1) Find the range of the data set { 15, 43, 29, 21, 14 }

Answer: 29

2) Find the area of the square with the side length of 11 cm.

Answer: 121 cm2

3) Find the Mean of the given data set: { 8, 16, 20, 26 }

Answer: 17.5

4) Find lateral surface area of cone with radius 8 cm and slant height 18 cm.

Answer: 452.16 cm2

Practice Divide Monomials

A polynomial is an algebraic expression with literal. Mostly we use x, y literal in polynomials. Literals are also called as variables. In polynomial, the variables have only positive integral exponents.

For example,  15 + 2x + x2 , 7x3 + 5xy2 + 12 y3 .

The first example is polynomials in one variable x.

The second example is polynomials in two variables x and y.

Polynomials having only one term are known as monomials. Monomial is one of the type of polynomials.  For example,

x3 , x2  x , y5, 5x5, 6y3 .

Let us practice division of a monomial problems.



Divide monomial problems for Practice:

There are two rules for dividing a monomial by a monomial. They are following,

Rule 1: The coefficient of the quotient of two monomials is equal to the quotient of the coefficients of the monomials in question.

Rule 2: In the quotient of two monomials, the variable part is equal to the quotient of the variable parts in the monomials in question.

Problem 1:

Divide by x5 by x3 monomials.

Solution:

x5 ÷ x3 = `x^5/x^3 `

= x2 .

Problem 2:

Divide by 15x5 by 5x4 monomials.

Solution:

15x5 ÷ 5x4 = `(15x^5)/(5x^4) `

= 5x .

Problem 3:

Divide by -20x4 by 10x monomials.

Solution:

- 20x4 ÷ 10x = `(-20x^4)/(10x) `

= `(-20/10)`` (x^4/x)`

= - 2 x3 .

Problem 4:

Divide by 3y3 by `sqrt(3)` y monomials.

Solution:

3y3 ÷ `sqrt(3)` y = `(3y^3)/(sqrt(3)y) `

= `(3/sqrt(3))`` (y^3/y)`

= `sqrt(3)` y2.

Problem 5:

Divide by 2x2 by 2x monomials.

Solution:

2x2 ÷ 2x      = `(2x^2)/(2x) `

= `(2/2)`` (x^2/x)`

=  x .

Problem 6:

Divide by -3x3 by x2 monomials.

Solution:

- 3x3 ÷ x2 = `(-3x^3)/(x^2) `

= `(-3/1)`` (x^3/x^2)`

= - 3 x .

Practice Problems on monomials:

Practice Problem 1:

Divide by `2/3` x2 by x monomials.

Answer:

`2/3` x .

Practice Problem 2:

Divide by `sqrt(5)` x4 by 5x3 monomials.

Answer:

`x/sqrt(5)` .

Practice Problem 3:

Divide by `sqrt(3)` a3 by 2a monomials.

Answer:

`(sqrt(3)a^2)/2` .

Practice Problem 4:

Divide by 4a4 by - 2 `sqrt(2)` a2 monomials.

Answer:

`-sqrt(2)a^2` .