Can one calculate area from a sloped square (as below)?

Question

A unit square, laid out with horizontal and vertical edges, is rotationally-symmetrically sectioned by diagonal segments from each corner to an opposite side such that a smaller interior square is formed by the four segments. (The term "rotationally-symmetrically means that if the square is rotated through 90 degree turns, the sectioned figure remains constant in appearance.). If the slope of the segment from the lower left-hand corner is c, what is an expression for the area of the interior square formed?

Answer 1

Please show a diagram.

Answer 2

Strategy: This is just algebra, albeit lengthy. Obtain equations and solve for points A and B, the topmost corners of the smaller square, from top-left to top-right.

Tactics: Point A is at the intersection of lines y = 0 + cx and y = 1 – (x/c) . These solve as:

1 – (x/c) = cx

xa = c/(c^2 + 1)

ya = c^2/(c^2 + 1)

Point B is at the intersection of the lines y = 1 – (x/c) and y = 1 – c(1-x) . These solve as:

1 – (x/c) = 1 + cx – c

xb = c^2/(c^2 + 1)

yb = 1 – c/(c^2 + 1)

Then r^2 (where r is the side of the smaller square) = (xb – xa)^2 + (yb – ya)^2 =

( (c^2/(c^2 + 1) – c/(c^2 + 1) )^2 + ( 1 – c/(c^2 + 1) - c^2/(c^2 + 1) )^2 =

(c^4 – 2c^3 + c^2)/(c^2 + 1)^2 + (1 – 2c + c^2) / (c^2 + 1)^2 =

(c^4 – 2c^3 + 2c^2 – 2c + 1) / (c^2 + 1)^2 .

Although this expression looks formidable, in calculations it's quick enough. Some notable values are:

c r^2

1.5 (1/13)

2 (1/5)

3 (2/5)

4 (153/189)

Curious how rapidly the rational expression becomes involved!