Square in square

This is not a really detailed solution, but by some consideration we can get a beautiful solution.

$\tan\theta=\dfrac13, \sec\theta=\sqrt{1+\tan^2\theta}=\dfrac{\sqrt{10}}{3}$

$\overline{BD}=10\cos\theta=10\dfrac{3}{\sqrt{10}}=3\sqrt{10}$

$\overline{BC}=\dfrac23\overline{BD}=2\sqrt{10}$

$\overline{BC}^2=(2\sqrt{10})^2=\boxed{40}$

I DO NOT KNOW HOW TO GENERATE A PICTURE!! I DO NOT THINK THAT IT IS ACCEPTABLE TO GIVE A PROOF WITHOUT ONE (I HAVE DONE THIS MANY TIMES!). WILL SOME-ONE TELL ME HOW TO GENERATE A PICTURE! Regards, David

Area=S^2 , Side Length Green Square = Area parallelogram \ its base(hypotenuse of right angled triangle) Since area \gram = b*h , since height is green square sidelength

Area 2 Right-Angled Triangles = 2 * 0.5 * 10*(10\3)=100\3 units^2

Area parallelogram = Area Purple Square - 2*Area right angled Triangle = 100-(100\3) = 200\3 units^2

base parallelogram = hypotenuse length from black point to yellow point = sqrt(10^2+(10\3)^2) =10 sqrt(10 \ 3) units Therefore, Sidelength=Area parallelogram\base parallelogram = 2 sqrt(10) unit Area green square = [2 sqrt(10)]^2 = 40 units^2

I solve this and similar problems (recent example (100 Day - day 7) with sides divided in 2) this way:

Observe that all the corresponding triangles are congruent.

Use the side division length as a unit and work out the areas inside in terms of that unit.

The area of a three part triangle on a side is 3/2 (3 units up, 1 at base, divided by 2).

There are two triangles inside that triangle that are similar. In this case with a ratio of 1:3 with area ratio 1:9.

The small triangles are one unit of area, the trapezoidal regions are 8 units.

The large three part triangle thus contains 10 units of area, so that the area of a small (one unit) triangle is (3/2) / 10 = 3/20.

There are 9 square units of area altogether. The sum of the areas outside of the inner square is 36 (3/20) = 5.4.

Thus inner square is 3.6 units and is 3.6/9 = .4 of the total unit area 9.

Multiplying the area of the given square by this proportion gives an area of 4. x 100 = 40.