Question

a regular pentagon shares a common center with a regular hexagon. if $overline{lm}paralleloverline{ab}$, across how many lines of reflection can the combined figure map back onto itself?

a. 6
b. 3
c. 1

Explanation:

Step1: Recall line - of - symmetry properties

A regular hexagon has 6 lines of symmetry. A regular pentagon has 5 lines of symmetry. For the combined figure to map onto itself, the line of reflection must be a line of symmetry for both the pentagon and the hexagon simultaneously.

Step2: Analyze the common lines of symmetry

Since $\overline{LM}\parallel\overline{AB}$, the common lines of symmetry are the ones that pass through the center of both polygons and are also lines of symmetry for both. The number of common lines of symmetry is determined by the greatest - common - divisor of the number of lines of symmetry of the two polygons. The greatest - common - divisor of 5 and 6 is 1.

Answer:

C. 1