Construct line segments of lengths $a/b$ and $\sqrt {a}$

Question

Given: 3 lines (longest one a, medium one b and unit 1)

How can you construct line segments of lengths $a/b$, $\sqrt {a}$ and $a/n$ with $n$ a positive integer?

What do you mean by "construct"? What I am allowed to do? Also, what have you tried? — M. Winter, Feb 15 '18 at 16:00

score 3 · Answer 1 · answered Feb 15 '18 at 16:14

3

Here are ideas for the fractions $a/b$ and $a/n$:

answered Feb 15 '18 at 16:14

M. Winter

29,928

Thnx! And how do you construct the root of a? – WinstonCherf Feb 15 '18 at 16:41
@Elce Please search the site before posting a question. This was found by just googling the question. – M. Winter Feb 15 '18 at 17:03

Deepak · Answer 2 · 2018-02-15T16:34:21.697

1

To supplement the other answer, for $\sqrt a$, you can proceed as follows. Note that $(a+1)^2 - (a-1)^2 = 4a$.

So: Start by constructing a line segment $L$ of length $a-1$, then use one end $O$ as the centre of a circle $C$ of radius $a+1$. Construct a perpendicular to $L$, let this meet the circle $C$ at $P$. Join $P$ to $O$. $OP$ has length $a+1$. You now have a right triangle with hypotenuse $a+1$ and one cathetus $a-1$, so the other cathetus is $2\sqrt a$. Bisect that to get $\sqrt a$.

edited Feb 15 '18 at 16:34

answered Feb 15 '18 at 16:17

Deepak

26,801

uh, no? Those will have length $\sqrt{2}a$. – Wojowu Feb 15 '18 at 16:19
@Wojowu Fixed it. – Deepak Feb 15 '18 at 16:35

Construct line segments of lengths $a/b$ and $\sqrt {a}$

2 Answers2