Find the area of the shaded region as a fraction of the area of the entire regular pentagram.
Source: Julia Robinson Mathematics Festival
Triangles 1 through 6 are congruent to each other. Triangle 7 is congruent to triangle 8. These facts need to be proved.
Area of Pentagram
The pentagram consists of eight triangles labeled 1 through 8. The area of the pentagram equals
Area of Shaded Region
The shaded region consists of triangles 5, 7, 6, and 2. The area of the shaded region equals
We conclude that the area of the shaded region is one half of the area of the pentagram.