What's the fastest way to split $45-45-90$ triangle into two parts with equal area (using a straightedge and compass)?
I know a similar question has been asked before, but I'm asking how to do this using only a straightedge and a ruler (i.e. not measuring). Same as in the other question, the $90°$ angle needs be perpendicular with the "floor", otherwise this woud be a completly obvious question (you would just draw a straight line up from the $90°$).
Just to be clear, you need to split the $45-45-90$ triangle below into two equal parts without turning it (or using the obvious way described above), using only a straightedge and a compass:
I came up with a pretty quick way to split it.
My way
- Start by putting the compass on the length of the base
- Start to draw a circle upwards until you hit the edge of the long side
- Draw a line from that point (when you hit the edge) to the base (perpendicularly) and et voilà, you have just split the triangle into two parts with equal areas
To see why this is true, let the shorter sides be $1$ and therefore the longer side is $\sqrt{2}$. The new smaller triangle that's created has a hypotenuse of $1$, therefore the catheti (the legs) are $\frac{1}{\sqrt{2}}$, and so we get that the are of the triange is:
$$A = \frac{1}{2}(\frac{1}{\sqrt{2}})^2 = \frac{1}{4}$$
which is half the triangle's area, as desired.
Questions
- Is my way the fastest way you can do it (apart from the obvious way described in the start)?
- Does anybody know any other ways to do it (say maybe only with a straightedge)?