We saw that a vertex angle of a regular n-gon has measure degrees. If three regular pol...

We saw that a vertex angle of a regular n-gon has measure degrees. If three regular polygons surround a vertex, the three vertex angle measures must add to 360°. Therefore, if three regular polygons with a sides, b sides, and c sides surround a vertex we have the following.

Consider the possibility of creating a semiregular tiling of three regular polygons such that a = 5, b = 5, and c = 10. Notice that Consider the following figure, where m, n, and p represent the number of sides in the polygon.

a. Point A is surrounded by two regular pentagons and a regular decagon. If point B is surrounded similarly, what is n?

b. If point C is surrounded similarly, what is m?

c. If point D is surrounded similarly, what is p?

d. What is the arrangement around E? What can you conclude?

e. For every triple of numbers a, b, and c that satisfies the equation in problem 46, determine whether a regular a-gon, a regular b-gon, and a regular c-gon will form a semiregular tiling.

Problem 46: a. Suppose one of the polygons is an equilateral triangle, so a = 3. Find all values for b and c that satisfy the equation.

b. Suppose one of the polygons is a square, so a = 4. Find all values for b and c that satisfy the equation.

c. Suppose one of the polygons is a regular pentagon, so a = 5. Find all values for b and c that satisfy the equation.

d. Suppose one of the polygons is a regular hexagon, so a = 6. Find all values for b and c that satisfy the equation.