Question

1. a. list all the angles of rotation up to 360° that will carry the regular polygon onto itself.

b. how many lines of symmetry are there?

1. answer the following using δabc with coordinates: a: (1, −1), b: (3, −1), c: (1, −4)

a. list the coordinates of the image using the rule (x, y) → (− x, y) and graph the image.
a: ( , ),
b: ( , ),
c: ( , )
b. what transformation has occurred (be specific)? if translation, include directions and distances. if rotation, include the center of rotation, angle of rotation and direction. if reflection, include the line of reflection.

Explanation:

Question 14a

Step1: Determine the number of sides

The polygon is a regular hexagon, so it has \( n = 6 \) sides.

Step2: Calculate the angle of rotation

The formula for the angle of rotation that maps a regular \( n \)-sided polygon onto itself is \( \frac{360^\circ}{n} \). For \( n = 6 \), this is \( \frac{360^\circ}{6}=60^\circ \).

Step3: Find all angles up to \( 360^\circ \)

We find all multiples of \( 60^\circ \) up to \( 360^\circ \): \( 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ \). But since \( 360^\circ \) is a full rotation (same as the original), sometimes it's excluded, but the problem says "up to \( 360^\circ \)", so we include it. Wait, actually, for a regular hexagon, the rotational symmetries are \( 60^\circ k \) where \( k = 1,2,3,4,5,6 \), but \( k = 6 \) gives \( 360^\circ \). However, typically, the non - trivial ones plus the full rotation. But let's check: when \( k = 1 \), \( 60^\circ \); \( k = 2 \), \( 120^\circ \); \( k = 3 \), \( 180^\circ \); \( k = 4 \), \( 240^\circ \); \( k = 5 \), \( 300^\circ \); \( k = 6 \), \( 360^\circ \). But in terms of carrying the polygon onto itself, the angles are \( 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ \). But sometimes \( 360^\circ \) is considered the same as \( 0^\circ \), but the problem says "up to \( 360^\circ \)", so we list all multiples of \( 60^\circ \) from \( 60^\circ \) to \( 360^\circ \).

A regular hexagon has a number of lines of symmetry equal to the number of its sides. For a regular \( n \)-sided polygon, the number of lines of symmetry is \( n \). For a hexagon, \( n = 6 \).

Step1: Apply the rule to point A

Given point \( A:(1,-1) \) and the rule \( (x,y)\to(-x,y) \). Substitute \( x = 1 \) and \( y=-1 \) into the rule. We get \( A':(-1,-1) \).

Step2: Apply the rule to point B

Given point \( B:(3,-1) \). Using the rule \( (x,y)\to(-x,y) \), substitute \( x = 3 \) and \( y = - 1 \). We get \( B':(-3,-1) \).

Step3: Apply the rule to point C

Given point \( C:(1,-4) \). Using the rule \( (x,y)\to(-x,y) \), substitute \( x = 1 \) and \( y=-4 \). We get \( C':(-1,-4) \).

Answer:

The angles of rotation are \( 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ \)

Question 14b