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

$3\left (\text{area 1}\right )+1\left (\text{area 7}\right )$
Answer: $\frac{1}{2}$.