# Euler circuits

Write ideas for a one-page summary on each of the following three topics from your text:

1. Euler circuits, including their definition, recognition, how to create them, and how to prove they exist.

Describe Nearest Neighbor, Sorted Edges, and Kruskal's algorithms. Illustrate their use.

Describe the characteristics of Simplex Method and Alternative to the Simplex Method. Illustrate their use in solving a practical problem.

2. Find three Web sites that demonstrate a practical application of the use of Euler circuits. Write a five to six sentence description of each Web site. For more information on a Links Assignment and samples, please go here: Links Assignment

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Provided is a response, including websites and sources referenced.

Euler Circuit is a special circuit that passes along each edge exactly once. There is a theorem, the Euler theorem that allows one to know exactly when it is possible to find an Euler circuit on a graph. Nearest neighbor is when you pick a starting vertex and form a circuit which starts and ends at that vertex by always travelling along the edge coming out of your vertex that has the lowest weight, being careful to never visit a vertex twice until all have been visited once. This is a quick and easy method, even though it may not give the best possible answer. Sorted edges occur when you "grab" edges until you have a complete circuit. The edge grabbed will always be the one with the lowest weight that has not been grabbed yet, as long as grabbing it does not cause a circuit to form ...

#### Solution Summary

The solution assists with writing ideas for a summary on each of the given topics regarding Euler circuits.

9 geometry problems

1. Angles ABC and CBD are complementary and the measure of CBD is twice of the measure of ABC. Angles ABD and DBE are supplementary angles.

a) Draw a sketch illustrating these angles.

b) Determine the measure of ABC.

c) Determine the measure of DBE.

2. Form the converse of each of the following statements. State whether the converse is true or false.

a) If two angles are congruent, then the angles are equal in measure

b) If two angles are vertical angles, then they are congruent

c) If two angles are adjacent, then they have the same vertex.

d) If two angles are complementary, then the sum of their measures is 90.

e) If two lines are perpendicular to the same line, then they are parallel.

3. Two parallel lines are cut by a transversal.

a) Find the measures of the angles if a pair of interior angles on the same side of the transversal are represented by (5x-32) and (x + 8).

b) Find the measures of the angles if a pair of interior angles on the same side of the transversal have measures such that one angle is 4 times the measure of the other.

4. Find the measure of angle x in the figure below.

[see attachment for figure]

5. Sarah uses a bag to apply icing to a cake she just baked. The shape of the bag is a cone. It has a diameter of 3in. and a height of 6in. How much icing will the bag hold when it is full?

6. Mike wants to have wall-to-wall carpeting for two rooms. One room is 9 ft by 12 ft and the other room is 12 ft by 15 ft. How much will it cost him if the carpet costs $10.50 per square yard?

7. Find the area of the shaded region. The diameter of the smaller circle is 9 in. and the diameter of the larger circle is 12 in. Use 3.14 for pi.

[see attachment for figure]

8. Triangles ABE and ACD are similar triangles. Find the length of side AE.

[see attachment for figure]

9. Find a world map. Focus on the following bordering countries in Southeast Asia: Burma, Cambodia, Laos, Thailand, and Vietnam.

a) Represent the map of those countries as a graph. (Use the first letter of the name of the country to label the vertices of the graph)

b) Determine (state yes or no) whether the graph in part (a) has an Euler path. If yes, give one such Euler path.

c) Determine (state yes or no) whether the graph in part (a) has an Euler circuit. If yes, give one such Euler circuit.

Please see attached file.

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