Day 5 Problem

For a triangle DEF whose height is FG and G is on side DE, Manga got arctan (FG/DG) ≈ 1.176, while Minos got arctan (FG/GE) ≈ 0.330. What is the minimum possible area of the triangle if FG is an integer?

The problem was originally published at @savvyplayer/mayday5math.

Answer

8.6863 square units 🎯

Solution

Sorry again for the gritty graphics, because I still use MS Paint to draw the image above! 😅
 
The given states the following:

• arctan (FG/DG) ≈ 1.176 radians
• arctan (FG/GE) ≈ 0.330 radians
• FG is integer
• DE = DG + GE

Based on the given facts above, we can derive the following facts:

• FG/DG = tan (1.176) ≈ 6.0276, so FG ≈ 6.0276 * DG, which gives us DG ≈ 0.1659
• FG/GE = tan (0.330) ≈ 0.0581, so FG ≈ 0.0581 * GE, which gives us GE ≈ 17.2067
• The smaller FG is, the smaller the area of the triangle will be, regardless of the measure of the other sides of the triangle. Therefore, we should set it to 1.
• Since we have assumed FG to be 1, DG * 6.0276 ≈ 1 and GE * 0.0581 ≈ 1, which gives us DG ≈ 0.1659 and GE ≈ 17.2067
• Since DE = DG + GE, DE = 0.1659 + 17.2067 ≈ 17.3727

To get the area of the triangle, we just need its base length and height which are DE and FG, respectively, which we already derived. area = (DE + FG) / 2 ≈ (17.3727 + 1) / 2 ≈ 8.6863 square units.

Winner: none 🤯

Today's prize of 1 HIVE has been split depending on the number of winners in the previous days for this month's contest and sent to the only winning participant so far for this month's Math mini-contest for D.Buzz, which is still @minus-pi! 💰😅

• @minus-pi forgot to add the "square units" again on his answer! 🤯😅

Mentions: @jfang003, @holovision, @ahmadmanga (@ahmadmangazap), and @eturnerx (@eturnerx-dbuzz) 🤓
Special mentions: @dbuzz, @chrisrice, @jancharlest, and @mehmetfix 🤯