Question

1. a triangle has rotation symmetry that can take any of its vertices to any of its other vertices. select all conclusions that we can reach from this.

a. all sides of the triangle have the same length.
b. all angles of the triangle have the same measure.
c. all rotations take one half of the triangle to the other half of the triangle.
d. it is a right triangle.
e. none of the sides of the triangle have the same length.
f. none of the angles of the triangle have the same measure.
(from unit 1, lesson 16.)

1. select all the angles of rotation that produce symmetry for this flower.

a. 30
b. 45
c. 60
d. 90
e. 120
f. 135
g. 180
(from unit 1, lesson 16.)

Explanation:

Question 4

• A triangle with rotation symmetry that maps any vertex to another is an equilateral triangle.
• In an equilateral triangle, all sides are equal (by definition), and all angles are equal (\(60^{\circ}\) each, since the sum of angles in a triangle is \(180^{\circ}\) and \(180\div3 = 60\)).
• Rotation symmetry of order \(3\) (for an equilateral triangle) does not imply reflection symmetry (which would be needed for “all rotations take one - half of the triangle to the other half of the triangle”). A right - triangle does not have such a high - order rotation symmetry (a right - isosceles triangle has rotation symmetry of order \(2\), and a non - isosceles right - triangle has no rotation symmetry). If all sides were different (or all angles were different), there would be no non - trivial rotation symmetry.

• To find the angles of rotation symmetry, we use the formula \(\frac{360^{\circ}}{n}\), where \(n\) is the number of “identical parts” of the figure.
• Looking at the flower - like figure, assume it has \(8\) identical parts (by counting the “petal - like” or repeating elements). Then the angles of rotation symmetry are \(\frac{360}{8}=45^{\circ}\), \(2\times45 = 90^{\circ}\), \(3\times45=135^{\circ}\), \(4\times45 = 180^{\circ}\)

Answer:

A. All sides of the triangle have the same length.
B. All angles of the triangle have the same measure.

Question 5