Question

unit 1 test review

1. for each diagram, list all angles of rotational symmetry that are less than 360 degrees

(a) (b) (c) (d) (e)

1. aeiouy is a regular hexagon. use the diagram to answer the following questions:

a. find the image of u after a 240 - degree clockwise rotation around r.
b. find the image of (overline{io}) after a 120 - degree clockwise rotation around r.
c. what is the image of (overline{io}) after a reflection across (overline{eu}).

1. regular octagon abcdefgh is inscribed in a circle with center x.

a. what is the image of segment ab after a 135 - degree clockwise rotation around x?
b. what is the image of cd after a reflection across line bf?

Explanation:

1. For the first - part of question 1 (finding angles of rotational symmetry less than 360 degrees):

• Recall that for a regular polygon with \(n\) sides, the angles of rotational symmetry \(\theta=\frac{360}{n}k\), where \(k = 1,2,\cdots,n - 1\). For non - regular shapes, we check the angles at which the shape coincides with itself.
• (a) The first shape is a hexagon - like shape. For a regular hexagon, \(n = 6\), and the angles of rotational symmetry less than 360 degrees are \(\theta=\frac{360}{6}k\), \(k = 1,2,3,4,5\), so \(\theta=60^{\circ},120^{\circ},180^{\circ},240^{\circ},300^{\circ}\).
• (b) A non - symmetric triangle has no non - zero angles of rotational symmetry less than 360 degrees.
• (c) The third shape has an angle of rotational symmetry of 180 degrees.
• (d) For a three - pointed star - like shape (\(n = 3\)), the angles of rotational symmetry are \(\theta=\frac{360}{3}k\), \(k = 1,2\), so \(\theta = 120^{\circ},240^{\circ}\).
• (e) Analyze the shape's symmetries. It has an angle of 180 degrees of rotational symmetry.

2. For question 2a:

• In a regular hexagon AEIOUY with center R, a 240 - degree clockwise rotation. Since the central angle between consecutive vertices of a regular hexagon is \(\frac{360}{6}=60^{\circ}\), a 240 - degree clockwise rotation moves 4 vertices clockwise. Starting from U, the image of U is A.

3. For question 2b:

• The central angle between consecutive vertices of a regular hexagon is 60 degrees. A 120 - degree clockwise rotation moves 2 vertices clockwise. For the line - segment \(\overline{IO}\), its image is \(\overline{YU}\).

4. For question 2c:

• Reflecting \(\overline{IO}\) across \(\overline{EU}\). Consider the perpendicular bisector properties. The image of \(\overline{IO}\) is \(\overline{AY}\).

5. For question 4a:

• In a regular octagon ABCDEFGH with center X, the central angle between consecutive vertices is \(\frac{360}{8}=45^{\circ}\). A 135 - degree clockwise rotation moves 3 vertices clockwise. The image of segment AB is segment DE.

6. For question 4b:

• Reflecting segment CD across line BF. Consider the symmetry properties of the octagon. The image of CD is HG.

Answer:

1. (a) \(60^{\circ},120^{\circ},180^{\circ},240^{\circ},300^{\circ}\); (b) None; (c) \(180^{\circ}\); (d) \(120^{\circ},240^{\circ}\); (e) \(180^{\circ}\)
2. a. A; b. \(\overline{YU}\); c. \(\overline{AY}\)
3. a. DE; b. HG