Pentagons and The Golden Ratio



Construct a Regular Pentagon

-Construct a regular pentagon in the space above. -Draw line segments from one vertex to two other non-adjacent vertices to create a triangle in the interior of the pentagon. -Your picture should look like the picture below.

Question 1:

Look at your picture. Find the triangle that is like triangle EBC in the picture above. What do you notice about this triangle? How can you confirm or disprove your idea?

Finish the Construction

-Draw a line segment from each vertex on the original pentagon to every other vertex on the pentagon. -Your pentagon should look like the picture below.

Question 2:

How many triangles similar to the first triangle do you see in the picture now?

Question 3:

How can you confirm that all the triangles are similar? Do the math.

Question 4:

What is the ratio between the longer side of each triangle to the base of each triangle?

Question 5:

How does this pentagon and these triangles relate to the triangles and rectangles in our previous work?