Question

lesson 16 practice problems

1. for each figure, identify any angles of rotation that create symmetry.
2. 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.

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

a. 45 degrees
b. 90 degrees
c. 135 degrees
d. 180 degrees
e. 225 degrees
f. 270 degrees

Explanation:

Step1: Analyze the triangle with rotation symmetry

If a triangle has rotation symmetry that can take any of its vertices to any of its other vertices, it is an equilateral triangle. In an equilateral triangle, all sides are equal (by the definition of an equilateral triangle) and all angles are equal (each angle is \(60^{\circ}\) as the sum of angles in a triangle is \(180^{\circ}\) and \(180\div3 = 60\)). So, for option A, in an equilateral triangle, by the property of an equilateral triangle \(a=b = c\) (where \(a,b,c\) are side - lengths). For option B, using the angle - sum property of a triangle \(\angle A+\angle B+\angle C=180^{\circ}\), and since the triangle is equilateral \(\angle A=\angle B=\angle C = 60^{\circ}\). For option C, in an equilateral triangle, a rotation of \(180^{\circ}\) does not take one half of the triangle to the other half (a reflection would be more appropriate for that in some cases, but rotation of \(180^{\circ}\) about the centroid does not split the triangle into two congruent halves in the way described).

Step2: Analyze the flower for rotation symmetry

The flower has 8 - fold symmetry. The formula for the angle of rotation symmetry is \(\theta=\frac{360^{\circ}}{n}\), where \(n\) is the number of times the figure coincides with itself in a full \(360^{\circ}\) rotation. Here \(n = 8\), and the angles of rotation are \(k\times\frac{360^{\circ}}{8}\), where \(k = 1,2,\cdots,8\). When \(k = 1\), \(\theta=45^{\circ}\); when \(k = 2\), \(\theta = 90^{\circ}\); when \(k=3\), \(\theta=135^{\circ}\); when \(k = 4\), \(\theta=180^{\circ}\); when \(k=5\), \(\theta=225^{\circ}\); when \(k = 6\), \(\theta=270^{\circ}\); when \(k=7\), \(\theta=315^{\circ}\); when \(k = 8\), \(\theta=360^{\circ}\)

Answer:

1. A. All sides of the triangle have the same length, B. All angles of the triangle have the same measure
2. A. 45 degrees, B. 90 degrees, C. 135 degrees, D. 180 degrees, E. 225 degrees, F. 270 degrees