Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Percentages


Percent is another way of saying 'for every hundred'.

    Any division where the divisor is $100$ is a percentage. 
       For example: $ \cfrac {20}{100} = 20 \%$  or
                            $ \cfrac {45}{100} = 45 \%$

In essence saying that,
                            $ \cfrac {x}{100} = x \%$

Since any ratio can also be expressed as a division, it can also be represented as a percentage. 

For ex., a ratio of $\cfrac{1}{2}$ or $1:2$ can be converted to a percent. 
$ \cfrac {1}{2}$ $=$ $ \cfrac {1 * 50}{2 * 50} = \cfrac {50}{100} = 50$ Per Cent $=  50\%$ 


Expressing $x \%$ as a Fraction

Any percentage can be expressed as a decimal fraction by dividing the percentage by $100$. 

As $x \% = x$ out of $100 = \cfrac {x}{100}$
$75 \% = 75$ out of $100 = \cfrac {75}{100}$ or $0.75$


Expressing the Fraction $\cfrac{a}{b}$ as a Decimal and as a Percentage

Any fraction can be expressed as a decimal and any decimal fraction can be converted into percentage by multiplying it with $100$. 

$\cfrac {1}{2} = 0.5 = 50 \%$

$\cfrac {1}{5} = 0.2 = 20 \%$

$\cfrac {1}{3} = 0.33 = 33.33 \%$

Percentages are Relative

Any fraction can be expressed as a decimal and any decimal fraction can be converted into percentage by multiplying it with $100$. 

If you score 18 on a 20 points quiz, that would be an absolute value. However if you said, you scored a $90 \%$, one can assume that you got a $90$ points out of $100$. 
You need to remember that the actual score would really depend on the denominator. 
  • $90 \%$ on a quiz of $60$ points would indicate a numerator of 54. 
  • $90 \%$ on a quiz of $20$ points would indicate a numerator of 18. 
  • $90 \%$ on a quiz of $70$ points would indicate a numerator of 63. 

Percentage Point

Difference between two percentage values. It is not equal to either percentage increase or percentage decrease. 

If we have 2 different absolute values, there are different ways of looking at them. 

Let one value be greater than the other. 

$(a)$ One value as a percentage of the other.

  • Ex - $x$ is what per cent of $y$?           
Let $x = k\%$ of $y$ 
$x = \cfrac {k}{100}$ of $y$
$ \rightarrow k = \cfrac{x}{y} * 100$

  • Ex - $y$ is what percentage of $x$?           
Let $y = p\%$ of $x$ 
$ \rightarrow p = \cfrac{y}{x} * 100$

  • Ex - Find the number whose $30\%$ is $36$?           
Let the number be $x$ 
Given that $30\%$ of the number is $36$
$ \rightarrow 30\%$ of $x = 36 \rightarrow \cfrac{30}{100} * x = 36$
$ \rightarrow  x =  \cfrac{36 * 100}{30} \rightarrow x = 120$

Thus our $x$ here is $120$. 


$(b)$ By what per cent is the greater quantity more than the smaller?

$Percentage$ $Increase = \cfrac {Greater - Smaller}{Smaller} * 100 \%$

  • Ex - By what percent is $100$ more than $90$?           
$ \cfrac{(100-90}{90} (100\%) = 11\cfrac{1}{9}\%$
i.e., The amount of $100$ is $11\cfrac{1}{9}\%$ more than $90$. 


$(c)$ By what per cent is the smaller quantity less than the greater?

$Percentage$ $Decrease = \cfrac {Greater - Smaller}{Greater} * 100 \%$

  • Ex - By what percent is $35$ less than $40$?           
$ \Bigg(\cfrac {40-35}{40}\Bigg)  (100\%) = 12.5\%$


  • Ex - A solution of $150$ litres contains $60\%$ of milk and the rest is water. How much water must be added to the above solution such that the resulting mixture contains $50\%$ of water (in litres)?           
Given: Solution $=150 l$
Quantity of milk $=60\%$ of solution
                         $=\cfrac{60}{100} * 150 = 90 l$
Thus quantity of water $=150 - 90 = 60 l$

Let the quantity of water to be added be $x$ litres. 
         
Thus $ \cfrac {(60 + x) \text{ liters} }{(150 + x) \text{ liters}} * 100 = 50$
$\rightarrow \cfrac{60 + x}{150 + x} = \cfrac {50}{100} $
$\rightarrow \cfrac{60 + x}{150 + x} = \cfrac {1}{2} $

$\rightarrow 120 + 2x = 150 + x $
$\rightarrow x = 30 $ liters


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