Question

The area of the right triangle in the diagram below is
A
square units.





image.png



What is the area of the square in the diagram in terms of
a
and
A
?


a
2
+
A
2
4
a
2


a
2
+
4
A
2
a
2


a
2
(
4
A
2
+
1
)


a
2
(
1
4
A
2
+
1
)

Answer 1

Let the legs of the right triangle be x and y and the hypotenuse be a. Then
- x^2 + y^2 = a^2,
- triangle area A = (1/2)xy, so xy = 2A.

A square placed in the right-angle corner has side s = xy/(x+y) (by similar triangles), so
s = 2A/(x+y) and the square's area is
s^2 = 4A^2/(x+y)^2.

But (x+y)^2 = x^2 + y^2 + 2xy = a^2 + 4A, hence
area(square) = 4A^2/(a^2 + 4A).

So the area of the square is 4A^2/(a^2 + 4A).