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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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Time and Work Problems

Key Facts If a person can finish a job in n days, then the work done by the person in 1 day is $\frac{1}{n}$th of the total job If a person completes $\frac{1}{n}$th of the total job in 1 day, then the time taken by the person to finish the complete job is n days.  Another version of work is pool/tank problems where there is an inlet of water and an outlet as well.   If an inlet fills a tank in n hours, then it fills $\frac{1}{n}$th part of the tank in 1 hour. i.e. work done by it in 1 hour is $\frac{1}{n}$ If an outlet empties a full tank in m hours, then it will empty $\frac{1}{m}$th part of the tank in 1 hour. i.e. work done by it is $-\frac{1}{m}$ The concept is not hard to understand, however the application can be extremely tricky. So it would only help to solve as many problems as you can on this subject. Let's look at some examples. Ex 1. A copy machine can copy a paper in $36$ minutes. If a second copy machine were to be used at the same ...
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Units Digit

Units digit of a number is the digit in the one's place of the number. For example, the units digit of $243$ is $3$. In competition math, you might come across occasional problems that ask you to find the units digit of an expression. These problems are more commonly found on Mathcounts Countdown Round. 1. The units digit of any number expressed as a power of $2$ repeats in the cycle $2, 4, 8, 6.$        $2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, ....$ etc. 2. The units digit of any number expressed as a power of $3$ repeats in the cycle $3, 9, 7, 1.$        $3^1 = 3, 3^2 = 9, 4^3 = 27, 3^4 = 81, 3^5 = 243, ....$ etc. 3. The units digit of any number expressed as a power of $4$ is $4$ if the power is odd and $6$ if the power is even.        $4^1 = 4, 4^2 = 16, 4^3=64, 4^4=256, ....$ etc. 4. The units digit of any number expressed as a power of $5$ is always $5$.        $5^1...
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More Numbers

Key Facts 1. Numbers of the form $\frac{p}{q}, q \neq 0$, where $p$ and $q$ are integers and those that can be expressed in the form of terminating or repeating decimals are called rational numbers .             Ex. $\frac{7}{32} = 0.21875, \frac{8}{15} = 0.5\bar{3}$ are rational numbers. 2. Properties of operations of rational numbers     For any rational numbers $a, b, c,$     (i) Rational numbers are closed under addition, multiplication, and subtraction.          $i.e., (a+b), (a-b),$ and $(a \cdot b)$ are all rational     (ii) Rational numbers follow the commutative law  of addition and multiplication.          $i.e., a + b = b + a$ and $a \cdot b = b \cdot a.$     (iii) Rational numbers follow the associative law of addition and multiplication,          $i.e., (a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = ...
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Popular posts from this blog

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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Irrational Numbers

A number which cannot be written in the form $ \cfrac {p}{q}$  where $p$ and $q$ are integers and $ q $  $ \neq 0$. Example: $ \sqrt {2}$, $ \sqrt {3}$, $ \sqrt {6}$, $ \sqrt {7}$, $ \sqrt {8}$, $ \sqrt {10}$ Theorem: If $p$ divides $x^3$, then $p$ divides $x$, where $x$ is a positive integer and $p$ is a prime number.  Proof :  Let $x = p_{1} p_{2}...p_{n}$ where $p_{1}, p_{2}, p_{3}.....p_{n}$  are primes, not necessarily distinct.  $ \rightarrow x^3 = p_{1}^3 p_{2}^3.....p_{n}^3$ Given that $p$ divides $x^3$ By fundamental theorem, $p$ is one of the primes of $x^3$.  By the uniqueness of fundamental theorem, the distinct primes of $x^3$ are same as the distinct primes of $x$.  $ \rightarrow p$ divides $x$ Similarly if $p$ divides $x^2$, then $p$ divides $x$, where $p$ is a prime number and $x$ is a positive integer. Example: Prove that  $ \sqrt {2}$ is irrational.  Solution : Let us assume that  $ \sqrt ...
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Incenter/Excenter Lemma

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Given $\triangle ABC$ with incenter $I$, extend $AI$ to meet the circumcircle of $\triangle ABC$ at $L$. Then, reflect $I$ over $L$, and name this point $I_a$. Then:  $(i)$ $BICI_a$ is a cyclic quadrilateral with point $L$ as the center of the circumscribed circle.  $(ii)$ $BI_a$ and $CI_a$ bisect the exterior angles of $\angle B$ and $\angle C$,  respectively.  Proof $(i)$ Notice that the the question is equivalent to proving the distance from $L$ to points $B, I, C, I_a$ are equal. In other words, we need to show that  $$LB = LI = LC = LI_a$$ Firstly, it is obvious that $LI = LI_a$ because by definition, $I_a$ is the reflection of $I$ over $L$, and reflections preserve the length of the segment.  Now, we are left with proving that $LB=LI$, because similar calculations will provide $LI=LC$.  We see that most of our given information involves angles, so we focus on proving $\angle LBI = \angle LIB$. Firstly,  $$\angle LBC = \angl...
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