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Showing posts with the label Fundamental Principle of Counting

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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Counting

In this blog post, we will study the very  Fundamental Principles of Counting (i) Multiplication  If one operation can be performed in $m$ ways and corresponding to each way of performing the first operation, a second operation can be performed in $n$ ways then the two operations can be performed in $m \cdot n$ ways. In other words, if there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \cdot n$ ways of doing both.  Here the different jobs/operations are mutually inclusive. It implies that all the jobs are being done in succession. In this case we use the ' and ' operator to account for all scenarios, and remember ' and ' refers to multiplication. Example :  A student has to select a letter from vowels and another letter from consonants, then in how many ways can he make this selection? Solution : Out of $5$ vowels he can select one vowel in $5$ ways and out of $21$ consonants he can select one consonant i...
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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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More Coordinate Geometry

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We have looked at some basics of coordinate geometry. Today, let's look at some intermediate applications of this topic. 1. Shoelace Theorem The Shoelace Theorem is an algorithm used to find the area of a polygon in the coordinate plane when the coordinates are known. If the coordinates are $(x_1, y_1), (x_2, y_2) ... (x_n, y_n), $ $\text{Area}=\frac{1}{2}\Big[(x_1y_2 - x_2y_1)+(x_2y_3 - x_3y_2) + ... + (x_ny_1 - x_1y_n)\Big].$ 2. Centroid The centroid is the point where the three medians intersect. It is also sometimes called the center of gravity for the triangle. Note: A median of a triangle is the line segment joining the vertex to the midpoint of the opposite side. If $(x_1,y_1), (x_2, y_2),$ and $(x_3,y_3)$ are the vertices of a triangle, then the coordinates of its centroid are $$\Bigg(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \Bigg).$$ 3. Incenter The incenter is the point where the three angle bisectors intersect. Note: An angle bisector of a triangle...
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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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