Altitude Term in Math

Introduction about altitude term in math:

            Altitude or height term is defined based on the context in which it is used. As a general definition, the term altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. In this article we shall discus about altitude term based problems.




Triangle:


The total space inside the triangle is called as area of that triangle.



Formula to find Area:
Area of right angle triangle (A) =1/2 (length x height) square unit

                                               = 1/2 l x h square unit.

                        Here, the term height refers the altitude of the triangle.

Example problem:



A right angle triangle has length 5cm and altitude 13 cm. Find the area of that triangle.
Solution:

Given:

            Length (l) =5cm

            Altitude (h) =13cm

Formula:

Area of triangle = 1/2 (l x h) square unit.

                           = 1/2 (5 x 13)

                           = 1/2 (65)

                           =65/2

                           =32.5

Area of triangle = 32.5 cm2



A right angle triangle has length 7.5m and height 10 m. Find the area of that triangle.
Solution:

Given:

            Length (l) =7.5m

            Altitude (h) =10m

Formula:

Area of triangle = 1/2 (l x h) square unit.

                           = 1/2 (7.5 x 10)

                           = 1/2 (75)

                           =75/2

                           =37.5

Area of triangle = 37.5 cm2


Rhombus:


The altitude of rhombus is the distance between base and opposite side of the base.



Formulas:

Area of the rhombus (A) = b x a

 b – Base of rhombus.
                                  
 a – altitude of rhombus                                  

If two diagonal lengths are given:

Area of the rhombus (A) = (d1 x d2)/2

Example problems:

1.      The altitude and base of rhombus are 11 cm and 6cm respectively. Find are of rhombus.

Solution:

      Given:
Altitude of rhombus (a) = 11 cm
Base of rhombus (b) = 6 cm

Area of the rhombus (A) = b x a square units.

     = 11 x 6

     = 66

      Area of the rhombus (A) = 66 cm2
                 

2.      The altitude and base of rhombus are 14 cm and 10cm respectively. Find are of rhombus.

Solution:

      Given:

   Altitude of rhombus (a) = 14 cm

   Base of rhombus (b) = 10 cm

  Area of the rhombus (A) = b x a square units.

   = 14 x 10

   = 140

  Area of the rhombus (A) = 140 cm2

Grid Table Math

Grid Table Math:-

Grid table is also multiplication table is used for finding the product of two numbers.

The following is grid table for first 12 numbers.

0	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

          


Math Examples on multiplication grid chart:-


Math Example:- 1

To find the product of 1 and 1 look at 1 in the first row and 1 in the first column.Draw  vertical and horizontal lines respectively as shown in the figure.




The point at which the vertical and horizontal line intersects is the needed result.The product of 1 and 1 is 1.

Math Example:- 2

To find the product of  2 and 3 take 3 in the row and 2 in the column.Draw  vertical and horizontal lines respectively as shown in the figure.




The point at which the vertical and horizontal line intersects is the needed result.The product of 2 and 3 is 6.

Math Example:- 3

To find the product of  6 and 5 take 5  in the row and 6 in the column.Draw  vertical and horizontal lines respectively as shown in the figure.




The point at which the vertical and horizontal line intersects is the needed result.The product of 6 and 5 is 30.

Math Problems on Grid Chart:-


Math Problem 1:-

Find the product of  2 and 10 using the grid table shown above.

solution:-

The given two numbers are 2 and 10 we need to find the product of these two numbers.

To find the product from the grid table shown above take 2 in the row and 10 in the column make horizontal and vertical line from 2 and 10 the point at which they meet is the result for the question.



The vertical and horizontal lines meet at 20 so the answer for the given question is 20

Math Problem 2:-

Find the product of  3 and 9 using the grid table shown above.

solution:-

The given two numbers are 3 and 9 we need to find the product of these two numbers.

To find the product from the grid table shown above take 3 in the row and 9 in the column make horizontal and vertical line from 3 and 9 the point at which they meet is the result for the question.



The vertical and horizontal lines meet at 27 so the answer for the given question is 27

Function Chart for Math

Introduction to function chart for math

A chart is a graphical representation of the data, in which the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can be representing the tabular numeric data, functions or some kinds of qualitative structures.





Function chart for math Examples


Function chart for math Example 1:

Function chart for F(x) = x3 Plug in numbers for x and find values for y,

Substitute x=-2,-1, 0,1,2,3

F(x) = x3

Substitute x=-2

F(x) =-23

F(x) =-8

The ordered pair is (-2,-8)

Substitute x=-1

F(x) =-13

F(x) =-1

The ordered pair is (-1,-1)

Substitute x=0

F(x) =-03

F(x) =0

The ordered pair is (0,0)

Substitute x=1

F(x) =13

F(x) =1

The ordered pair is (1,1)

Substitute x=2

F(x) =23

F(x) =8

The ordered pair is (2,8)

Substitute x=3

F(x) =33

F(x) =27

The ordered pair is (3,27)

As we have done with the table below.

x    -2    -1    0    1    2    3
f(x)         -1    0    1    8    27


Math Function chart



Function chart for math Example 2:

Function chart for F(x) = x2 Plug in numbers for x and find values for y,

Substitute x=-2,-1, 0,1,2,3

F(x) = x2

Substitute x=-2

F(x) =-22

F(x) =4

The ordered pair is (-2,4)

Substitute x=-1

F(x) =-12

F(x) =1

The ordered pair is (-1,1)

Substitute x=0

F(x) =-02

F(x) =0

The ordered pair is (0,0)

Substitute x=1

F(x) =12

F(x) =1

The ordered pair is (1,1)

Substitute x=2

F(x) =22

F(x) =4

The ordered pair is (2,4)

Substitute x=3

F(x) =32

F(x) =9

The ordered pair is (3,9)

As we have done with the table below.

x    -2    -1    0    1    2    3
f(x)    4    1    0    1    4    9

Math Function chart



Function chart for math Example 3:

Function chart for F(x) = 6*x2 Plug in numbers for x and find values for y,

Substitute x=-2,-1, 0,1,2,3

F(x) =6* x2

Substitute x=-2

F(x) =6*-22

F(x) =24

The ordered pair is (-2,24)

Substitute x=-1

F(x) =6*-12

F(x) =6*1

The ordered pair is (-1,6)

Substitute x=0

F(x) =6*02

F(x) =0

The ordered pair is (0,0)

Substitute x=1

F(x) =6*12

F(x) =6

The ordered pair is (1,6)

Substitute x=2

F(x) =6*22

F(x) =24

The ordered pair is (2,24)

Substitute x=3

F(x) =6*32

F(x) =54

The ordered pair is (3,54)

As we have done with the table below.

x    -2    -1    0    1    2    3
f(x)    24    6    0    6    24    54



Math Function chart

Classifying Math Diagrams

Introduction to classifying math diagrams:

In mathematics we use different type of diagrams. In math, diagrams are classified as per the shape size of the figure. There are different types of diagrams are used in mathematics. In mathematics, diagrams are mainly used in geometrical problem. The diagrams in mathematics gives the clear explanation for particular problem. Classification of diagrams are given below,

• Circle
• Square
• Rectangle
• Rhombus
• Triangle
• Ellipse
• Parallelogram

Classifying math diagrams - Explanation
Classifying math diagrams - Circle:

• Circle has the radius (R), and diameter (d).
• The origin of the circle is indicated as O.
• The Interior angle of the circle is 360°.
• Area of the circle is pr2.
• Circumference of the circle is 2pr.

Classifying math diagrams - Square:


        

• Square has four equal sides.
• The side length of the square is denoted as S.
• Total angle of the square is 360°
• All the four sides have the same angle 90°
• Square has two diagonals and both the diagonals are equal in length
• Area of the square is s2

Classifying math diagrams - Rectangle:

• Rectangle have the four side lengths
• Length and width of the rectangle is denoted as L and W
• Opposite side of the rectangles are equal in length and also have same angle.
• Rectangle has two diagonal length
• Total angle of the rectangle is 360°
• Each side length of the rectangle  has 90°
• Area of the rectangle is (length * width)

Classifying math diagrams - Triangle

• A triangle has three side lengths.
• The sum of angle of the triangle is 180°
• Triangles are classified based on its side length and angle
• In right triangle, one angle should be 90°
• In equilateral triangle, all the three side lengths are equal
• Area of the triangle is (1 / 2) * b * h

Classifying math diagrams - Rhombus:

         

• Rhombus has four equal side length.
• It is also called as diamond.
• It is one of the quadrilateral diagram
• Opposite angles of the rhombus is equal
• Diagonals are perpendicular to each other
• Area of the rhombus is (base * height)

Surd Online

Introduction to surd online

Expressions like `sqrt(16)` ,`root(3)(27)` ,etc. have exact numerical value.They are terminating and rational numbers.These numbers are perfect roots (square root and cube root). But expressions such as `sqrt(2)`,`root(3)(3)` , etc.cannot be written as exact numeric value. Such numbers are called irrational and it is convenient to leave them in the form as in Decimal form  they would go on non-terminating. These are called surds. Surds are numbers of the form `root(n)(k)` ,where k is not a perfect nth power of any number.`root(n)(k)` is a surd of nth order.

Ex 1: `sqrt(2)` is a surd of 2nd order.

`root(3)(3)`is a surd of 3rd order.

`root(4)(5)`is a surd of 4th order

A surd has infinite number of non-terminating decimals.Surds are always irrational.

Ex 2: Which of the following numbers are surds?

`sqrt(7)`
`sqrt(9)`
`root(3)(27)`
`sqrt(5)`
`sqrt(100)`
`root(4)(16)`
Solution :

`sqrt(7)` is a surd.
`sqrt(9)` is not a surd because `sqrt(9)` = 3
`root(3)(27)` is not a surd because `root(3)(27)` = 3
`sqrt(5)` is a surd.
`sqrt(100)` is not a surd because `sqrt(100)` = 10
`root(4)(16)` is not a surd because `root(4)(16)` = 2


Simplifying a surd:

Consider the entire surd `sqrt(18)`  , it has 9 as one of its factors which is a perfect square.So `sqrt(18)` can be expressed as a product of rational number and a surd.

`sqrt(18)`  = `sqrt( 9 * 2)`

= `sqrt(9)` *`sqrt(2)`

= 3 *`sqrt(2)`

= 3`sqrt(2)`  This is called a mixed surd.

Ex 3: Express the mixed surd 4`sqrt(2)` as an entire surd.

Solution:  4`sqrt(2)` =`sqrt(16)`  * `sqrt(2)`       because 4 = `sqrt(16)`

= `sqrt( 16 * 2)`

= `sqrt(32)`


Addition , Subtraction and Multiplication on surd online


Addition and subtraction of surds is simple, however operation can be performed only on surds of same order and the radical must be same.That is number inside the root must be the same.Lets consider an example:

Ex 1: 5`sqrt(11)` + 2`sqrt(11)`  = `sqrt(11)` ( 5 + 2)  = 7`sqrt(11)`

Ex 2: 6`sqrt(5)` - 3`sqrt(5)` = `sqrt(5)` ( 6 - 3) = 3`sqrt(5)`

Ex 3: Simplify 6`sqrt(3)` + `sqrt(75)`

Solution : First simplify `sqrt(75)` .Express it as mixed surd.

`sqrt(75)` = `sqrt(25 * 3)` = 5`sqrt(3)`

Then,    6`sqrt(3)` + 5`sqrt(3)` = 11`sqrt(3)`

Multiplication of similar surds gives rational number.  i.e.   `sqrt(x)` * `sqrt(x)` = x

Ex 4: Simplify (`sqrt(6)`)2

Solution: (`sqrt(6)` )2 = `sqrt(6)` * `sqrt(6)`

= `sqrt(6 * 6)`

=  6

Multiplication of unlike surds give irrational number.

Ex 5: Simplify `sqrt(8)` * `sqrt(3)`

Solution: `sqrt(8)` * `sqrt(3)` = `sqrt(8 * 3)`

= `sqrt(24)`

= `sqrt(4 * 6)`

= 2`sqrt(6)`

Ex 6: Simplify 3`sqrt(5)` * 5`sqrt(3)`

Solution:

Method 1:    3`sqrt(5)` * 5`sqrt(3)` = `sqrt(3 * 3 * 5)` * `sqrt(5 * 5 * 3)`

= `sqrt(3 * 3 * 5 * 5 * 5 * 3)`

= 3 * 5 `sqrt(15)`

= 15`sqrt(15)`

Method 2:  3`sqrt(5)` * 5`sqrt(3)` = 3 * 5 *`sqrt(5)` * `sqrt(3)`

= 15`sqrt(5 * 3)`

= 15`sqrt(15)`

Ex 7: Simplify 2`sqrt(18)` * 3`sqrt(20)`

Solution: 2`sqrt(18)` * 3`sqrt(20)` = 2`sqrt(9 * 2)` * 3`sqrt(4 * 5)`

= 2*3`sqrt(2)` * 3*2`sqrt(5)`

= 6*6*`sqrt(2)` * `sqrt(5)`

= 36`sqrt(2*5)`

= 36`sqrt(10)`


Rationalisation of the denomiator on surd online


We express a radical fraction, such as `sqrt(5)` /`sqrt(2)` in a form that has rational denominator.

Ex 1: Simplify `sqrt(5)` / `sqrt(2)`

Solution: In order to rationalise we multiply numerator and denominator by `sqrt(2)`

`(sqrt(5))/(sqrt(2))`  * `(sqrt(2))/(sqrt(2))`= `(sqrt(5 * 2))/(2)`  = `(sqrt(10))/(2)`

Ex 2: simplify `(sqrt(35))/(sqrt(15))`

Solution: `(sqrt(35))/(sqrt(15))` * `(sqrt(15))/(sqrt(15))`    = `(sqrt(7 * 5 * 5 * 3))/(sqrt(15 * 15))`   = `(5sqrt(21))/(15)`   = `(sqrt(21))/(3)`

Consider (2 + `sqrt(3)`) * ( 2 - `sqrt(3)`)  = 4 - 3 = 1. Their product is a rational number. Hence 2 - `sqrt(3)` is called the conjugate of the surd 2+`sqrt(3)` or vice verse.`sqrt(a)` + `sqrt(b)` is called the conjugate of `sqrt(a)` - `sqrt(b)` if their product is a rational number

Ex 3 : Simplify `(1)/(sqrt(2) + sqrt(3))`

Solution: Here we have to rationalise the denominator.Hence multiply and divide numerator and denominator by `sqrt(2)`-`sqrt(3)`(the opposite sign of the denominator)

We have, `(1)/(sqrt(2) + sqrt(3))` *  `(sqrt(2) - sqrt(3))/(sqrt(2) - sqrt(3))` = `(sqrt(2)- sqrt(3))/(2-3)` = `sqrt(3)` - `sqrt(2)`

Study Online Dilation

Study Online Dilation Introduction:

Dilation is a change (notation Dk) that produces a picture that is the same shape as the original, but is a different size. Dilation stretches or shrinks the original diagram.

The explanation of dilation contains the ratio or scale factor and the middle of the dilation. The middle of dilation is a set point in the plane about which every point are expanded or contracted.  It is the just invariant point under dilation.

A dilation of scalar factor k whose middle of dilation is the basis  written by:  Dk (x, y) = (kx, ky). If the scale factor, k, is larger than 1, the picture is an enlargement.

If the scale factor is 0 to 1, the picture is a reduction.


Study Online Dilation - Definition:


A dilation is a vary of the plane, Dk, such that if O is a set point, k is a non-zero real number, and P' is the picture of point P, then O, P and P' are collinear and `(OP ' )/(OP)` = k.
Notation:  Dk(x, y) = (kx, ky )

Examples for Study online Dilation:


Study online Dilation - Example 1:

PROBLEM:

Sketch the dilation picture of triangle ABC with the middle of dilation at the origin and a scale factor of 2.

Examine: Notice how EACH coordinate of the triangle has been multiplied by the scale factor (x2).

Study online Dilation - Example 2:

PROBLEM:

Sketch the dilation picture of pentagon ABCDE with the middle of dilation at the origin and a scale factor of 1/3.



Examine: Notice how EACH coordinate of the pentagon has been multiply the scale factor (1/3).

Note: Multiplying by 1/3 is the same as dividing by 3!

Study online Dilation - Example 3:

PROBLEM:

Sketch the dilation diagram of rectangle EFGH with the middle of dilation at point E and a scale factor of 1/2.



Examine:

E and its picture are the same.  It is main to observe the distance from the middle of the dilation, E, to the other points of the diagram.  Notice EF = 6 and E'F' = 3.

Note:

Be sure to measure distances for this problem.

Geometric Word Problems : 2

Introduction to geometric word problems:     

Geometry is a theoretical subject, but easy to learn, and it has many real practical applications. Finally, geometry has developed into a skillfully arranged and sensibly organized body of knowledge.

Geometry gives the planning of different geometrical shapes and figures in our daily life such as articles in the houses, wells, buildings, bridges etc. The term 'Geometry' means a study (learn) of properties of figures and shapes and the relationship between them.

Example problems of geometric word problems:


Geometric word problem 1:

Find the largest possible rectangular area we can enclose, assuming we have 144 centimeters of fencing. What is the implication of the dimensions of this largest possible enclosure?

Geometric word problem Solution:

Let the length be L and the width be W. We have 144 centimeters of fencing, so the perimeter equation is:

2L + 2W = 144

Dividing by 2 to make things simpler, then we get

L + W = 72

Area of the rectangle formula as,

A = L × W

We can substitute for either one of the above variables by solving the perimeter equation:

L + W = 72
L = 72 – W   

Then we get,

A = (72 – W) × W   

 = 72W – W 2

This equation is in the format of ax2+bx+c.



A = –W 2 + 72W

The vertex of a parabola is the point (h, k), where h = –b/2a.  In this case:

h = –(72)/(2×(–1)) = 36

To find the "k" part of the vertex, all we do is plug 36 in for W:

k = –(36)2 + 72(36) = 3888                                         

The largest possible area is 3888 square centimeters

Now w e can find the length by using the value of width. Then we get,

L = 72– W = 72 – 36 = 36

Then the length and width are the same: 36 centimeters.

Therefore, the largest possible rectangular area is in the shape of a square.

Geometric word problem 2:

A square has an area of twenty five square centimeters. What is the length of each of its sides?

Geometric word problem Solution:

The formula for the area A of a square with side-length ‘a’ is:

A = a2

Substitute the value of A in the above formula: Then we get,

25= a2

√25 = a
5 = a

After re-reading,

  a=5 cm

The length of each side is 5 centimeters.


Practice geometric word problems:


A circle has an area of 81π square units. What is the length of the circle's diameter?
 Answer: 9

A piece of 16-gauge copper wire 54 cm long is twisted into the shape of a rectangle whose width is twice its length. Find the dimensions of the rectangle.
 Answer: L=9 cm and W=18 cm