# Show constructibility

• September 25th 2010, 01:58 PM
courteous
Show constructibility
"Constructible lengths are those that can be built from known lengths by the operations of $+$, -, $\times$, $\%$ and $\sqrt$."

Quote:

Show, using similar triangles, that if lengths $l_1$ and $l_2$ are constructible, then so are $l_1l_2$ and $l_1/l_2$.
How do you go about it? (Blush)

How do you get a proper minus sign in $LaTeX$? I've tried "-", "$-$", "--", ... nothing works.
• September 26th 2010, 10:02 AM
BobP
Construct a triangle $PQR$ with $PQ$ of length $l_{1}, QR$ of length $1$ and $RQ$ of any length at all (so long as $PQR$ is a triangle).
Now construct a line $Q'R'$, parallel to $QR$ and of length $l_{2},$ followed by lines $P'Q'$ and $P'R'$ parallel to there none dash equivalents.
The two triangles will be similar and if you work out the length of $P'Q'$, it will be seen to be $l_{1}l_{2}.$
A similar routine can be used to construct a line of length $l_{1}/l_{2}.$