A polygon is called convex if every line segment from one vertex to another lies entirely...

A polygon is called convex if every line segment from one vertex to another lies entirely within the polygon. To triangulate a polygon, we take some of these line segments, which don’t cross one another, and use them to divide the polygon into triangles. Prove, by strong induction for all naturals n with n ≥ 3, that every convex polygon with n sides has a triangulation, and that every triangulation contains exactly n − 2 triangles. (Hint: When you divide an n-gon with a single line segment, you create an i-gon and a j-gon for some naturals i and j. What does your strong inductive hypothesis tell you about triangulations of these polygons?)

Expert Solution

Summary: Thus, by the principle of Strong Induction, we have proven the statement to be asked to prove in question. The hint given the question was very useful in the solution.

Colby Messinger answered 7 months ago

A polygon is called convex if every line segment from one vertex to another lies entirely...

A polygon is called convex if every line segment from one vertex to another lies entirely within the polygon. To triangulate a polygon, we take some of these line segments, which don’t cross one another, and use them to divide the polygon into triangles. Prove, by strong induction for all naturals n with n ≥ 3, that every convex polygon with n sides has a trian-gulation, and that every triangulation contains exactly n − 2 triangles. (Hint: When you divide...

(Area of a convex polygon) A polygon is convex if it contains any line segment that...

(Area of a convex polygon) A polygon is convex if it contains any line segment that connects two points of the polygon. Write a program that prompts the user to enter the number of points in a convex polygon, then enter the points clockwise, and display the area of the polygon. Sample Run Enter the number of points: 7 Enter the coordinates of the points: -12 0 -8.5 10 0 11.4 5.5 7.8 6 -5.5 0 -7 -3.5 -5.5 The...

Complete exercise 9.20 from the text. (a) Create a function called polygon that draws a polygon...

Complete exercise 9.20 from the text. (a) Create a function called polygon that draws a polygon in a polar plot. Your function should have a single input parameter – the number of sides. (b) Use a for loop to create a figure with four subplots…You should use the function you created in part (a) to draw each polygon. Use the index parameter from the for loop to specify the subplot in which each polygon is drawn, and in an expression...

Suppose ?⃗ (?,?)=−??⃗ +??⃗ and ? is the line segment from point ?=(2,0) to ?=(0,3). (a)...

Suppose ?⃗ (?,?)=−??⃗ +??⃗ and ? is the line segment from point ?=(2,0) to ?=(0,3). (a) Find a vector parametric equation ?⃗ (?) for the line segment ? so that points ? and ? correspond to ?=0 and ?=1, respectively. ?⃗ (?)= (b) Using the parametrization in part (a), the line integral of ?⃗ along ? is ∫??⃗ ⋅??⃗ =∫???⃗ (?⃗ (?))⋅?⃗ ′(?)??=∫?? ?? with limits of integration ?= and ?= (c) Evaluate the line integral in part (b). (d)...

for every bounded, closed and convex subset K of Y, there exists a retract from Y...

for every bounded, closed and convex subset K of Y, there exists a retract from Y onto K prove the lemma

Evaluate the line integral along the given paths. xy ds (a) C: line segment from (0,0)...

Evaluate the line integral along the given paths. xy ds (a) C: line segment from (0,0) to (5, 4) C: counterclockwise around the triangle with vertices (0, 0), (8, 0), and (0, 2)

Evaluate the line integral along the given paths. xy ds C (a) C: line segment from...

Evaluate the line integral along the given paths. xy ds C (a) C: line segment from (0, 0) to (7, 4) counterclockwise around the triangle with vertices (0, 0), (8, 0), and (0, 4)

Write a C++ function called parse that reads one line of user input from the keyboard...

Write a C++ function called parse that reads one line of user input from the keyboard and creates an array of the strings found in the input. Your function should be passed the array and a reference variable that is to be assigned the length of the array. Prompt the user and read the input from within the function. For example: If the user inputs copy this that, the resulting array would have length 3 and contain the strings “copy”, “this”, and “that”....

At one time or another, almost every retailer claims that their first and only priority is...

At one time or another, almost every retailer claims that their first and only priority is to make the customer happy. Few, however, can duplicate what Nordstrom does. For the Seattle-based upscale retailer, “the customer is always right” is not just a promotional motto but a way of life that guides the organization. The entire company is directed towards one goal––catering to the needs of customers. Sales associates are given incredible freedom to do whatever is needed to make customers...

At one time or another, almost every retailer claims that their first and only priority is...

At one time or another, almost every retailer claims that their first and only priority is to make the customer happy. Few, however, can duplicate what Nordstrom does. For the Seattle-based upscale retailer, “the customer is always right” is not just a promotional motto but a way of life that guides the organization. The entire company is directed towards one goal––catering to the needs of customers. Sales associates are given incredible freedom to do whatever is needed to make customers...

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A polygon is called convex if every line segment from one vertex to another lies entirely...

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A polygon is called convex if every line segment from one vertex to another lies entirely...

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