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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Coordinate Geometry

Coordinate Geometry is the study of geometry using algebraic tools like equations, relations, and operations. Before we start our deep dive, let's go over some basics of coordinate geometry.

A coordinate grid consists of 2 perpendicular lines, which are called axes, and they are labeled like a number line would be. The horizontal axis is known as the x-axis, while the vertical axis is called the y-axis. The point of intersection of the two axes is known as the origin. The coordinate of the origin will always be $(0, 0)$. The numbers along the axes are used to locate points in the plane. The point at which a graph intersects the x-axis is known as the x - intercept and the point of intersection with the y-axis is known as the y - intercept.

The coordinates also known as an Ordered Pair, is a set of values that expresses the distance a point lies from the origin. It is written in the format $(x, y)$ where $x$ is the value on the x-axis and the $y$ is the value on the y-axis.


The intersection of X-axis and Y-axis divides the coordinate plane into 4 quadrants. In Quadrant I, both the x-coordinate and y-coordinate are positive. Quadrant II has the x-coordinate as negative, and the y-coordinate as positive. In Quadrant III, both x and y-coordinates are negative, and Quadrant IV has the x-coordinate as positive but the y-coordinate negative. 


Midpoint Formula
For calculating the midpoint coordinates of a line segment on the coordinates plane, here is what you would use $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$ where the endpoints of the line segment are $(x_1, y_1)$ and $(x_2,y_2)$.

We can further extend this concept if we were to find the coordinates of the point dividing the line segment joining the points $(x_1,y_1)$ and $(x_2,y_2)$ internally, in the ratio $m : n$ using the below formula. Let me however warn you that this is more of a intermediate concept than basic math. 
$$\Big(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\Big)$$

Distance Formula
To find out the distance between 2 points $A(x_1, y_1)$ and $B(x_2,y_2)$ you would use the formula $\sqrt{(x_2-x_2)^2+(y_2-y_1)^2}$.

For example, the distance between the points $(6, -4)$ and $(3,0)$ is 
$$\sqrt{(3-6)^2+(0-(-4))^2} = \sqrt{9+16} = 5 \text{ units}$$

Slope of a Line
Next, let us understand the Slope of a Line. 
Slope is the measure of the steepness of a line. A line can have positive, negative, zero(horizontal) or undefined(vertical) slope. Can be calculated using "rise over run" or more explicitly, $\frac{y_2-y_1}{x_2-x_1}$.
  • A line with a negative slope falls from left to right (its left side is higher than its right), and its slope is less than 0.
  • A line with a positive slope rises from left to right (its right side is higher than its left), and its slope is greater than 0.
  • A horizontal line has a slope of 0; it neither rises nor falls and is parallel to the x-axis.
  • The slope of a vertical line is undefined because you don’t know whether it’s rising or falling; it has no slope and is parallel to the y-axis



The Three Forms for the Equation of a Line
A linear equation is an algebraic representation of a straight line. There are 3 forms of a linear equation:



Let's look at some problem solving strategies to apply when asked to find the equation of a line.




Slopes of Parallel and Perpendicular Lines
  • If two lines are parallel, then the value of their slopes must be the same. 
  • If two lines are perpendicular, then the product of their slopes must be equal to $-1$. In other words, the two slopes will be opposite reciprocals of each other. An example of opposite reciprocals would be $3$ and $-\frac{1}{3}$.


Distance of a Point from a Line
The length of the perpendicular from a point $(x_1,y_1)$ to the line $ax+by+c=0$ is $$\Bigg|\frac{ax_1+by_1+c}{\sqrt{a^2+b^2}}\Bigg|$$

Coordinate Geometry is a very important topic, so make sure you take your time to thoroughly understand the content reviewed. If you have any questions, do not hesitate to reach out. I can provide additional resources and problems for practice. Enjoy. 

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Equilateral and Equiangular Polygons

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