B5. In a square $A B C D$ with side length 2, we draw a line from each corner to the midpoint of each of the two opposite sides. For example, we connect $A$ to the midpoint of $B C$ and to the midpoint of $C D$. The eight lines we thus draw together bound an octagon within the square (see the figure).

What is the area of this octagon?

![](https://cdn.mathpix.com/cropped/2024_04_17_d68bed7f77e1173b56f6g-2.jpg?height=377&width=377&top_left_y=425&top_left_x=1476)

## C-questions

For the C-questions, the answer is not the only thing that matters; there should also be a reasoning that shows your answer is correct. Each correctly worked-out C-question is worth 10 points. Partial solutions can also earn points. Therefore, write everything clearly and submit your scratch paper as well.

NOTE: Solve each C-question on a separate sheet of paper and also submit the corresponding scratch paper per question.
5. In the figure, the upper half of the square is shown. Triangle PST and triangle $P S Q$ are mirror images of each other in line $P C$. Each has an area equal to one-eighth of the octagon. Triangle $Q R S$ has the same area as $P S Q$ because both triangles have a base of $\frac{1}{2}$ and they have the same height. The three triangles together form the gray triangle in the figure with a base of 1 and a height of $\frac{1}{2}$. The gray triangle thus has an area of $\frac{1}{2} \cdot 1 \cdot \frac{1}{2}=\frac{1}{4}$, and each of the three small triangles has an area of $\frac{1}{3} \cdot \frac{1}{4}=\frac{1}{12}$. The area of the octagon is therefore $8 \cdot \frac{1}{12}=\frac{8}{12}=\frac{2}{3}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_5922b08c02b8c45ce4e3g-2.jpg?height=497&width=888&top_left_y=571&top_left_x=595)

## C-questions