CPM Homework Help : GC Problem 12-77

Find the area of the shaded region of the regular pentagon at right. Show all work.

A pentagon with the vertices labeled B, C, M, A, and N starting at the lower left vertex and going counter clockwise. B, C, is 4. Two diagonals are drawn: B, A, and C, A. The triangle A, B, C, is shaded.

Hint:

Label the shaded region as $ΔABC$ .

More Help 1:

The measure of each angle in a pentagon is equal to $\frac{180(5 \cdot 2)}{5} = 180^\circ$ .

Therefore, the measure of $\angle BAC$ is equal to $\frac{108}{3} = 36^\circ$

More Help 2:

Divide $\Delta ABC$ into two triangles. Use a trigonometric ratio to solve for $AD$ .

Step:

Solve for the length of AD: $\tan 18^\circ = \frac{2}{AD}$

Hint:

Once you know the length of AD, use the formula (1/2)(Base)(Height) to find the area of the shaded triangle.

From the vertex, A, is dropped a line perpendicular to the base, B, C, at point, D, dividing the base into 2, 2 unit segments. Right triangles A, B, D and A, D, C are created.