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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...

Right Triangles

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Let's take a look at right triangles and some of their special properties today. As a quick refresher, a right triangle is a triangle with a right angle. We can call the side opposite the right angle as the  hypotenuse , and the two sides adjacent to the 90^{\circ} angle as the  legs . In the image to the right, AC is the hypotenuse, and the two legs are AB and BC. Pythagorean Theorem One of the most famous and useful theorems in geometry is the Pythagorean Theorem. The theorem states that in right triangle \triangle ABC with hypotenuse c and legs a and b, a^2+b^2=c^2
Lets apply this in a problem to see how it works.  Problem 1 : Let right triangle \triangle ABC with right angle at B have hypotenuse of r+1, and legs of length 7 and r. Find r.  Solution:  We know that the sum of the squares of the legs is equal to the square of the hypotenuse. In other words, we have 7^2 + r^2 = (r+1)^2.
Simplifying this a...

Angles

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An angle is formed when two rays share an origin. There are many different types of angles formed when two lines/rays/line-segments intersect. The point of origin/intersection is called the  vertex , and the two rays are called the  sides . When we name an angle, we always put the vertex in the middle. For example, in the image on the right, \angle ABC is an angle with vertex at point B. Angles: All Different Kinds A 90^{\circ} angle is a Right Angle . Lines, segments, or rays that form a right angle are said to be Perpendicular . An angle smaller than 90^{\circ} is an Acute Angle . An angle between 90^{\circ} and 180^{\circ} is an Obtuse Angle . An angle that measures 180^{\circ} is a Straight Angle . An angle of more than 180^{\circ} is a Reflex Angle . Two angles that add up to 180^{\circ} are known as Supplementary Angles . Two angles that add up to 90^{\circ} are known as Complementary Angles . When two lines intersect, they ...

Sides and Angles of a Triangle

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Lets look at some basic facts related to sides and angles in triangles. For any triangle, the sum of any two sides will always be larger than the third side. We could also say that the difference of lengths of any two sides must be less than the length of the third side. (Triangle Inequality Theorem) The sum of the three interior angles of a triangle is 180^{\circ}. The sum of all exterior angles of an n-sided polygon is 360^{\circ}. The two base angles of an Isosceles triangle are congruent. The sum of all interior angles of an n-sided polygon is (n − 2) \cdot 180^{\circ} . An exterior angle of a triangle is equal to the sum of the two opposite interior angles. (Exterior Angle Theorem) For a triangle, the opposite side of a bigger interior angle is longer than that of a smaller angle, and vice versa. In a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. (Pythagorean Theorem) Here are some problems that you would solve using...

Number of Trailing 0's

Today, we will be looking at problems that ask you to find the number of trailing 0's in any factorial. Say for example, we are asked to find the number of trailing zeroes of 101! Reminder: n! = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1 Simply put, trailing zeroes are zeroes at the end of the number without any non zero digits to the right of them. Ex. - there are 4 trailing zeroes in the number 20340000. Trailing 0's are formed when a multiple of 2 is multiplied by a multiple of 5. So, all we have to do is count the number of 5's and 2's in 101!. Let's start with counting the number of 5's. The numbers 5, 10, 15, 20, 25, ... 95, 100 all contribute one factor of 5 to the factorial, so we have 20 factors of 5. However, some numbers have more than one multiple of 5. For example, the numbers 25 = 5 \cdot 5 50 = 2 \cdot 5 \cdot 5  75 = 3 \cdot 5 \cdot 5 100 = 4 \cdot 5 \cdot 5   all have an ex...

Train Problems

The key to solving moving train problems is to understand the distance that the needs to be covered or the relative speeds at which the objects are commuting in the problems. Lets take a look at both of these cases and talk about how to solve these types of questions.  If the train is passing a Stationary object, think Distance  - If the problem asks for the time taken by a moving train to pass a pole or standing man or anything similar that basically is a point, you will need to find the time taken by the train to cover the length of the train itself.   - If the problem asks for the time taken by a moving train to pass a bridge or a tunnel or anything that has a length of itself, then the time taken by the train to pass that object will be the length of the train + the length of the object. If the train is passing a Moving object, think Speed  - If the problem involves 2 moving objects, you would usually want to calculate the net speed betwee...

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