a If we draw diagonals between opposite vertices, we create 12 congruent triangles.
B
b What fraction is left if 9 of 12 pieces have been eaten?
A
a About 403.2 cm^2
B
b About 101 cm^2
Practice makes perfect
a Let's first illustrate the top of the cookie.
If we draw diagonals between opposite vertices of the dodecagon, we get 12 congruent triangles. We can claim congruence by the SSS (Side-Side-Side) Congruence Theorem. This is because the diagonals bisect each other and because the dodecagon is regular, which means its 12 sides are congruent.
If we find one triangle's area we can then calculate the area of the dodecagon. In order to do this, we will first determine the sum of the dodecagon's interior angles.
The sum of the interior angles are 1800^(∘). Since this is a regular dodecagon, each interior angle will measure 1800^(∘)12=150^(∘). Additionally, the diagonal between two opposite vertices bisects the angles at each vertex. Therefore, if each of the dodecagon's angles is 150^(∘), half of this is 75^(∘).
If we draw the height from the vertex angle it cuts the opposite side in two equal halves.
By finding the height of this triangle we can determine its area and then the area of the dodecagon. With the given information, we can find the height by using the tangent ratio.
Now we can calculate the area of the triangle and then the dodecagon by multiplying this number by 12.
Area: (1/2(6)(11.20))12≈ 403.2 cm^2
b As explained in Part A, the dodecagon can be divided into 12 congruent triangles. If 9 of the 12 pieces have been eaten, we have 312 of the cookie left. By multiplying the area we found in Part A by this fraction, we can determine what area of the cookie is left.
