Regular Pentagram

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
6\left (\text{area 1}\right )+2\left (\text{area 7}\right )

Area of Shaded Region
The shaded region consists of triangles 5, 7, 6, and 2. The area of the shaded region equals
3\left (\text{area 1}\right )+1\left (\text{area 7}\right )

We conclude that the area of the shaded region is one half of the area of the pentagram.

Answer: \frac{1}{2}.

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About mvtrinh

Retired high school math teacher.
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