a. Geometry How many sides does the polygon have? How many of its diagonals come from one vertex

step 1 All right, so this is a geometry connection. So this is something that you will potentially lear

A. Geometry How many sides does the polygon have? How many...

Key Concepts

Polygon

A polygon is a closed two-dimensional figure composed of a finite number of straight line segments connected to form a closed chain or circuit. The study of polygons involves understanding the relationships between their sides, angles, and other geometric properties.

Polygon Sides

The number of sides in a polygon is a fundamental property that determines many of its other characteristics, including the number of vertices and the possible diagonals that can be drawn. It is also essential in deriving formulas related to perimeter, area, and the sum of interior angles.

Diagonals from a Vertex

A diagonal is a line segment connecting two non-adjacent vertices in a polygon. From any given vertex in an n-sided polygon, the number of diagonals is determined by excluding the vertex itself and its two adjacent vertices, leading to the formula n - 3 for the number of diagonals from one vertex.

Total Number of Diagonals

The total number of diagonals in a polygon can be calculated by multiplying the number of diagonals from one vertex by the number of vertices, and then recognizing that each diagonal is counted twice (once from each end). This results in the formula n(n - 3)/2, which is pivotal for understanding the relationship between the polygon's sides and its diagonals.

Standard Form of a Polynomial

Writing a polynomial in standard form involves arranging its terms in descending order based on the degree of each term. This process helps in simplifying, comparing, and performing operations like addition, subtraction, and factoring on polynomial expressions.

Transcript

00:01 All right, so this is a geometry connection.

00:03 So this is something that you will potentially learn next year or in future sections of this book.

00:11 But they're making this connection that there is a relationship between the number of sides of any polygon, any multi -sided figure, and the number of diagonals that you can draw, either from one of the vertices, as you can see here, or from all of the vertices.

00:28 And so that would look like this, and then from the next one, and from the next one, and so on.

00:41 All right, so let's begin with what we see.

00:44 I counts that there are one, two, three, four, five, six sides in this polygon, which makes it a hexagon.

00:53 And the number of diagonals that are shown here, starting at one vertex and going across, this side is not a diagonal.

01:01 So this is number one, two, three, and that's it.

01:07 There are three diagonals that can go from one of the vertices to each of the vertices that are not adjacent.

01:18 All right, next.

01:20 A polygon with n sides has how many diagonals from one vertex.

01:29 So something that i noticed about this, is that the number of diagonals is three less than the number of sides, because it can't connect to itself and it also cannot connect to, it's two adjacent vertices.

01:58 And so that means automatically you have three fewer diagonals than the number of vertices that there actually are, or number of sides.

02:11 And so if there's n sides, and there would be n minus three diagonals.

02:19 So now we're given a relationship that the number of diagonals for all the vertices, so not just from the one that is shown here, but from every single diagonal, every single vertex has all of its diagonals drawn, and some of them will overlap, meaning you wouldn't recounts the one that starts at this vertex and goes back to our first one.

02:40 That's the same as the original.

02:43 So there are n over 2 times n minus 3 vertices...

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