For the construction we have the following diagram.
Notice that from the above we obtain the following triangle.
As we can see, ∠ A is common for △ ABC and △ ADE. Additionally, the sides of △ ABC that include ∠ A are proportional to the sides of △ ADE that include ∠ A.
AD/AB = 40/100 = 2/5 =
40/100 =AE/AC
Therefore, △ ADE ~ △ ABC because of the Side-Angle-Side (SAS) Similarity Theorem. This allows us to write the following proportion.
DE/BC = AD/AB = 2/5 [0.2cm]
⇕
DE = 2/5 BC
We have proven that DE is two-fifths the length of BC.
Extra
Extra
The sector compass works because, as seen before, it constructs a pair of similar triangles whose scale factor depends on where the marks are drawn.
