Find the area of the shaded region as a fraction of the area of the entire regular pentagram.

Source: Julia Robinson Mathematics Festival

**SOLUTION**

Congruent Triangles

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.

**Answer**: .

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Retired high school math teacher.