Question

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. (lesson 1 - 16) 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.

Explanation:

Step1: Recall rotation - symmetry concept

A triangle has rotation - symmetry if it can be rotated about a point and map onto itself. For an equilateral triangle (a special case of a triangle), all sides are of equal length and all angles are equal (60°). When we rotate an equilateral triangle by 120° or 240° about its centroid, it maps onto itself. But for a general triangle, this is not the case.

Step2: Analyze each option

• Option A: Only equilateral triangles have all sides of the same length. General triangles do not, so this is not a general property for all triangles with rotation - symmetry.
• Option B: Only equilateral triangles have all angles of the same measure. General triangles do not, so this is not a general property for all triangles with rotation - symmetry.
• Option C: Rotations do not necessarily divide a triangle into two equal - halves. For example, a non - isosceles non - equilateral triangle with some non - trivial rotation symmetry (if exists) will not have this property.
• Option D: A right - triangle does not have any special relation with rotation symmetry in general.
• Option E: Triangles with rotation symmetry do not have all sides of the same length in general.
• Option F: Triangles with rotation symmetry do not have all angles of the same measure in general.

Since none of the above statements are true for all triangles with rotation symmetry, the answer is none of the above.

Answer:

F. None of the angles of the triangle have the same measure.
E. None of the sides of the triangle have the same length.