Lesson 6 Symmetries of Regular Polygons Solidify Understanding

Ready

1.

What is the smallest fraction of a full circle that the wagon wheel needs to turn in order to appear the very same as it does now? How many degrees of rotation would that be?

A circle divided into 8 pieces.

2.

What is the smallest fraction of a full circle that the propeller needs to turn in order to appear the very same as it does right now? How many degrees of rotation would that be?

A point with five lines drawn evenly around the point like a propeller

3.

What is the smallest fraction of a full circle that the Ferris wheel needs to turn in order to appear the very same as it does right now? How many degrees of rotation would that be?

A circle divided into 18 pieces made to look like a ferris wheel.

Set

4.

Draw the lines of symmetry for each regular polygon. Fill in the table, including an expression for the number of lines of symmetry in an -sided polygon.

Number

of Sides

Number

of Lines

of Symmetry

An equilateral triangle, square, regular pentagon, regular hexagon, regular heptagon, regular octagon

5.

Draw all of the diagonals in each regular polygon. Fill in the table and find a pattern. Is the pattern linear, exponential, or neither? How do you know? Attempt to find an expression for the number of diagonals in an -sided polygon.

Number

of Sides

Number

of Diagonals

An equilateral triangle, square, regular pentagon, regular hexagon, regular heptagon, regular octagon

6.

Find the angle(s) of rotation that will carry the -sided polygon below onto itself.

A regular dodecagon

7.

What are the angles of rotation for a -gon? How many lines of symmetry (lines of reflection) will it have?

8.

What are the angles of rotation for a -gon? How many line of symmetry (lines of reflection) will it have?

9.

A regular polygon has rotational symmetry for an angle of . How many sides does it have? Explain.

10.

A regular polygon has rotation symmetry for an angle of . How many sides does it have? How many lines of symmetry does it have?

Go

Use tools to make your work precise.

11.

Reflect point over the line of reflection and label the image .

A coordinate plane with x- and y- axis of 1-unit increments with a line with y-intercept 3 and slope of 3. Point A is located at (1,-4). x–5–5–5555y–5–5–5555000line of reflection

12.

Reflect point over the line of reflection and label the image .

A coordinate plane with x- and y- axis of 1-unit increments with a line with y-intercept -4 and slope of -4. Point A is located at (-6,-4). x–5–5–5555y–5–5–5555000line of reflection

13.

Reflect triangle over the line of reflection and label the image .

A coordinate plane with x- and y- axis of 1-unit increments with a line with y-intercept 4 and slope of 1/2. Triangle with vertices located at A(-1,-5), B(-2,-1), and C(1,-3). x–5–5–5555y–5–5–5555000line of reflection

14.

Reflect parallelogram over the line of reflection and label the image .

A coordinate plane with x- and y- axis of 1-unit increments with a line with y-intercept 4 and slope of 1/2. Parallelogram with vertices located at A(6,5), B(5,8), C(8,7), D(9,4) x–10–10–10–5–5–5555101010y–10–10–10–5–5–5555101010000line of reflection

15.

Given triangle and its image , draw the line of reflection that was used.

A coordinate plane with x- and y- axis of 1-unit increments. Triangle with vertices located at X(-5,3), Y(-4,6), and Z(-6,7). Triangle with vertices located at X'(-2,0), Y' (1,1), and Z'(2,-1) x–10–10–10–5–5–5555y–5–5–5555101010000

16.

Given parallelogram and its image , draw the line of reflection that was used.

A coordinate plane with x- and y- axis of 1-unit increments. One parallelogram with vertices Q(-5,2), T(-3,3), S(-4,6), R(-6,5). Another parallelogram with vertices Q'(3,2), T'(1,3), R'(4,5), S'(2,6). x–5–5–5555y–5–5–5555000