# Right-angled triangles with rectangles inside

Infinitely many rectangles BDEF can be inscribed within the right-angled triangle ACE, with the point B belonging to the segment AC, the point F belonging to the segment AE and the point D belonging to the segment EC. The length of AE is 1 unit and the length of EC is 2 units.

C
|\
D-B
| |\
E-F-A

1. Draw 2 possible triangles BDEF (I already did that).
2. Explain why the triangles ACE and ABF are similar (I already did that).
3. If EF = $x$ units show that BF= $2(1-x)$ units. (same)
4. Find in terms of $x$ the area of the rectangle BDEF (I got it to $5x^2-8x+4$)
5. Find $x$ such that the area of the rectangle is maximized.
6. Find the ratio between the maximum area of the rectangle BDEF and the area of the triangle ACE.
• (4) can't be right. It should be $2x(1-x)$. And are you asking for just (5) and (6)? – Parcly Taxel May 12 '17 at 15:50

$x(1-x)$ is maximum for $x=1/2$. That follows, for instance, from AM-GM inequality. Or from $y=x(1-x)$ being the equation of a parabola with vertex $(1/2,1/4)$.