Answer:
2/9
Explanation:
You want the area of a square inscribed in an isosceles right triangle with legs 1 unit.
Ratios
Referring to the attached figure, we have right triangle ABC with sides AB = AC = 1. The ratio the side length to the hypotenuse of an isosceles right triangle is 1 : √2, so we have ...
FG/FA = √2/1 ⇒ FG = FA·√2
FH/FB = 1/√2 ⇒ FH = FB/√2
Since FG=FH, we have ...
FA·√2 = FB/√2
FB = 2·FA
Side length
AB = 1 = FB +FA = (2FA) +FA = 3FA
This means FA = 1/3 and the side length of the square is √2/3.
The area of the square is ...
A = s² = (√2/3)² = 2/9
The area of the square is 2/9 square units.
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Additional comments
The centroid of the square is the same as the centroid of the triangle. We constructed the figure using the triangle centroid to help divide the sides into thirds.
The figure can be divided into 9 congruent triangles of the size of ∆AGF. The square is comprised of 4 of those.