Each symmetry in D 3 .

Question

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Answer 1

Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 1, Problem 1E

To determine

To Describe:Each symmetry in D₃.

Expert Solution & Answer

Explanation of Solution

Given: An equilateral triangle.

Calculation:

Three rotations at 0, 120 and 240 gives the symmetric in an equilateral triangle and also flips over three different axes (vertex) at the midpoint of the opposite side gives the symmetric in an equilateral triangle. So total there are six symmetries in equilateral triangle.

The three rotations at 0, 120 and 240 which gives the symmetric are:

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 1

The three symmetries obtain by flips over three different axes are -

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 2

Thus, there are six symmetries in equilateral triangle.

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Answer 2

Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 1, Problem 1E

To determine

To Describe:Each symmetry in D₃.

Expert Solution & Answer

Explanation of Solution

Given: An equilateral triangle.

Calculation:

Three rotations at 0, 120 and 240 gives the symmetric in an equilateral triangle and also flips over three different axes (vertex) at the midpoint of the opposite side gives the symmetric in an equilateral triangle. So total there are six symmetries in equilateral triangle.

The three rotations at 0, 120 and 240 which gives the symmetric are:

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 1

The three symmetries obtain by flips over three different axes are -

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 2

Thus, there are six symmetries in equilateral triangle.

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Answer 3

Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 1, Problem 1E

To determine

To Describe:Each symmetry in D₃.

Expert Solution & Answer

Explanation of Solution

Given: An equilateral triangle.

Calculation:

Three rotations at 0, 120 and 240 gives the symmetric in an equilateral triangle and also flips over three different axes (vertex) at the midpoint of the opposite side gives the symmetric in an equilateral triangle. So total there are six symmetries in equilateral triangle.

The three rotations at 0, 120 and 240 which gives the symmetric are:

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 1

The three symmetries obtain by flips over three different axes are -

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 2

Thus, there are six symmetries in equilateral triangle.

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Answer 4

Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 1, Problem 1E

To determine

To Describe:Each symmetry in D₃.

Expert Solution & Answer

Explanation of Solution

Given: An equilateral triangle.

Calculation:

Three rotations at 0, 120 and 240 gives the symmetric in an equilateral triangle and also flips over three different axes (vertex) at the midpoint of the opposite side gives the symmetric in an equilateral triangle. So total there are six symmetries in equilateral triangle.

The three rotations at 0, 120 and 240 which gives the symmetric are:

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 1

The three symmetries obtain by flips over three different axes are -

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 2

Thus, there are six symmetries in equilateral triangle.

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Answer 5

Chapter 1, Problem 1E

To determine

To Describe:Each symmetry in D₃.

Expert Solution & Answer

Explanation of Solution

Given: An equilateral triangle.

Calculation:

Three rotations at 0, 120 and 240 gives the symmetric in an equilateral triangle and also flips over three different axes (vertex) at the midpoint of the opposite side gives the symmetric in an equilateral triangle. So total there are six symmetries in equilateral triangle.

The three rotations at 0, 120 and 240 which gives the symmetric are:

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 1

The three symmetries obtain by flips over three different axes are -

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 2

Thus, there are six symmetries in equilateral triangle.

Answer 6

Chapter 1, Problem 1E

To determine

To Describe:Each symmetry in D₃.

Expert Solution & Answer

Explanation of Solution

Given: An equilateral triangle.

Calculation:

Three rotations at 0, 120 and 240 gives the symmetric in an equilateral triangle and also flips over three different axes (vertex) at the midpoint of the opposite side gives the symmetric in an equilateral triangle. So total there are six symmetries in equilateral triangle.

The three rotations at 0, 120 and 240 which gives the symmetric are:

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 1

The three symmetries obtain by flips over three different axes are -

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 2

Thus, there are six symmetries in equilateral triangle.

Answer 7

Chapter 1, Problem 1E

To determine

To Describe:Each symmetry in D₃.

Answer 8

Expert Solution & Answer

Explanation of Solution

Given: An equilateral triangle.

Calculation:

Three rotations at 0, 120 and 240 gives the symmetric in an equilateral triangle and also flips over three different axes (vertex) at the midpoint of the opposite side gives the symmetric in an equilateral triangle. So total there are six symmetries in equilateral triangle.

The three rotations at 0, 120 and 240 which gives the symmetric are:

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 1

The three symmetries obtain by flips over three different axes are -

Contemporary Abstract Algebra, Chapter 1, Problem 1E , additional homework tip 2

Thus, there are six symmetries in equilateral triangle.