3 sets of sides of the hexagon in the figure faced each other are parallel.
As for each of 3 sets, the ratio of the length of a short side and a long side is 1 : 3.
Find the area ratio of the area of a shadow area, and the area of a hexagon.
As for each of 3 sets, the ratio of the length of a short side and a long side is 1 : 3.
Find the area ratio of the area of a shadow area, and the area of a hexagon.
Answer
13 : 22
Solution
As shown in Fig. 1, three triangles are made on the outside of a hexagon by drawing an auxiliary line and three triangles are congruent.
△ADE and △ABC are homothetic and a homothetic ratio is DE : BC = 1 : (3 + 1 + 1) = 1 : 5.
An area ratio is 1 × 1 : 5 × 5 = 1 : 25.
If the area of △ADE is set to 1, △ABC is 25.
Comparing the area of △ADE and △DEH, since FG = AE, AE : EH = 1 : 3 and an area ratio is 1 : 3.
The area of △DEH is 3.
In the same way, the area of △FGD and △HGI is also 3.
Therefore, the area of a shadow area is 25 - (1 + 3) × 3 = 13.
The area of a hexagon is 25 - 1 × 3 = 22.
