Geometry

What is it? It’s basically questions regarding size, shape, position, and properties. Used everywhere

and anywhere.

Table of Contents

• Properties of Tangents by Rolando

• Exploring Polygons by Akhilesh

• Angles and Arcs of a Circle by Jillian

Properties of Tangents

•  The tangent function is one of the circular transcendental functions. It finds use in solving plane right triangle problems and in most problems which involve periodic properties. Its definition is in terms of the sine and cosine functions. 

• This is the tangent function.

• It is a periodic function which repeats itself every 2p radians. An important property to remember is that tan(0)=0.

• Properties to remember.

Exploring Polygons

• A polygon is a plane figure formed by 3, or more segmen

• It can be either convex or concave.

So question for you.

If you answered….

• Any of the ones outlined in red, you were right!

• The simplest polygon is Triangle, and all of the terms of a triangle apply to it. Such as…

Number of Sides Polygon3 Triangle4 Quadrilateral5 Pentagon6 Hexagon7 Heptagon8 Octagon9 Nonagon

A polygon is determined by its number of sides.

Another example.•

To solve this, it’s really sim

ple. Com

bine like terms

and solve for x.• The perimeter of a

Pentagon is 40cm, solve for X.

• The total angle of a polygon can be determined by the following formula.

• 180(n-2). Where n, is the number of sides in the polygon.

• Examples:• A triangle has 3 sides. Plug it into the

formula.• 180(n-2)• 180(3-2)• 180(1)• • 180°. Therefore, the answer is 180°.•

• The last thing I want to mention is that there are a few formulas to find sides and the area of a polygon. Some of them will be shown in the next slide.

Solving for Sides.

a=inradius, n is the number of sides, and π is pi. And tan is tangent, which is a function calculated in radians.

r=radius, n is the number of sides, and π is pi, and sin is sine a function calculated in radians.

Solving for Area

Where, S=is the length of any side. N=is the number of sides, π= is pi, and tan is a function calculated in radians.

Where, r=radius, n=number of sides, π= pi, and sine is a function calculated in radians.

Solving for Area

A=apothem(the perpendicular distance of a side from the center), N=number of sides,

and π=pi.

A=apothem, and P is the perimeter. Now, you may wonder how to determine what the perimeter is. This is fairly simple.

P=(n)(s)P is Perimeter. N is the number of sides, and S is the length of one side. AND THAT IS IT FOR

POLYGONS.

Angles and Arcs of a Circle

• There are several different angles associated with circles.

• Central angles are angles formed by any two radii in a circle. The vertex is the center of the circle.

• An arc of a circle is a continuous portion of the circle. It consists of two endpoints and all the points on the circle between these endpoints.

Arcs• Semicircle: an arc whose endpoints are the

endpoints of a diameter. It is named using three points. The first and third points are the endpoints of the diameter, and the middle point is any point of the arc between the endpoints.

• Minor arc: an arc that is less than a semicircle. A minor arc is named by using only the two endpoints of the arc.

• Major arc: an arc that is more than a semicircle. It is named by three points. The first and third are the endpoints, and the middle point is any point on the arc between the endpoints.

Measuring Arcs.• Degree measure of a semicircle: This is

180°. Its unit length is half of the circumference of the circle.

• Degree measure of a minor arc: Defined as the same as the measure of its corresponding central angle. Its unit length is a portion of the circumference. Its length is always less than half of the circumference.

• Degree measure of a major arc: This is 360° minus the degree measure of the minor arc that has the same endpoints as the major arc. Its unit length is a portion of the circumference and is always more than half of the circumference.

AXB Is a semicircle, so it’s length would be half the circumference.

• The following theorems about arcs and central angles are easily proven.

• Theorem 68: In a circle, if two central angles have equal measures, then their corresponding minor arcs have equal measures.

• Theorem 69: In a circle, if two minor arcs have equal measures, then their corresponding central angles have equal measures

The End.