Why is BD?

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[Originally posted in the Discussions]
Why is BD = (2 root 3/2 root 3 + 4)x2?

It's great that you are trying to understand this step! It's a little confusing, because many steps were done at the same time.
Let's look at a different example of a simpler ratio:
Like in the Day 13: Your Turn question, the purple angles are equal, and the line going through the triangle is an angle bisector. The difference is that the lengths of the two sides which make up the bisected angle are simpler: 1 and 2. These sides are in the ratio 1 : 2, which we can illustrate with a pie chart:
The blue section corresponds to the "1" out of the total "1 + 2", so it takes up 1/3 not 1/2 of the whole pie. We would find that the segment labeled with a "?" takes up 1/3 of the length of x.
Now, back to our problem: Instead of 1 and 2, the lengths of the sides that touch the bisected angle are 2 x root(3) and 4, so the segment that we want, BD, takes up
$$ \frac{2\sqrt{3}}{2\sqrt{3} + 4} $$
of the length of the third triangle side. The length of BD is this ratio of the length of the third side, or$$ \frac{2 \sqrt{3}}{2\sqrt{3} + 4} \times \text{ length of third side} $$
which equals:
$$ \frac{2\sqrt{3}}{2\sqrt{3} + 4} \times 2 $$
is because the total length of the third side (BD + CD) is equal to 2.
I hope this helps! It's a pleasure to help you with this, and congratulations on almost finishing the course. We really hope you have enjoyed learning with this problembased approach and enjoy tackling challenges more and more.
Happy Learning!
The Daily Challenge Team