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Showing posts with the label Geometric Sequences

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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Progressions

Let us observe the following pattern of numbers.     $(i)$   $5, 11, 17, 23, ............$    $(ii)$   $6, 12, 24, 48, ............$   $(iii)$   $4, 2, 0, -2,-4, ............$   $(iv)$   $\cfrac {2}{3}, \cfrac{4}{9}, \cfrac{8}{27}, \cfrac {16}{81},......... $ In example, $(i)$, every number (except 5) is formed by adding $6$ to the previous numbers. Hence a specific pattern is followed in the arrangement of these numbers. Similarly, in example $(ii)$, every number is obtained by multiplying the previous number by $2$. Similar cases are followed in examples $(iii)$ and $(iv)$.  SEQUENCE A systematic arrangement of numbers according to a given rule is called a sequence.     The numbers in a sequence are called its terms. We refer the first term of a sequence as $T_{1}$, the second term as  $T_{2}$ and so on. The $n$th terms of a sequence is denoted by  $T_{n}$, which may also be...
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Geometric Sequences

Let's take a look at geometric sequences today. In a way, they are similar to Arithmetic Sequences , but just think common ratio instead of common difference. Let's begin by taking a look at the geometric sequence $2, 6, 18, 54...$. The initial term , or the first term is $2$. The common ratio  is defined as the number you multiply to each term to get the next one. In this case, the common ratio would be $3$, because $2 \cdot 3 = 6, 6 \cdot 3 = 18, 18 \cdot 3 = 54$, and so on. Common problems include finding the $n$th term, or finding the sum of such sequences. To find the $n$th term of a geometric sequence, we use the formula: $$a_n = ar^{n-1}$$ Where $a_n =$ the $n$th term in the sequence $a =$ the initial term $r =$ the common ratio Your Turn! $1.$  Find the $7$th term in the sequence $5, 10, 20, 40...$ (Scroll down to the bottom of the page for the answer) Next, let's take a look at how to find the sum of such a sequence. Let's find the sum of the seque...
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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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