1. ## Contstructing Segments

Given three segments whose lengths are 1, a, and b, construct segments of length a + b, |ab|, ab, $\displaystyle \frac{a}{b}$, and $\displaystyle \sqrt{ab}$. These are Euclidean constructions.

2. Hello, ReneePatt!

Surely, you can do the first two . . .

Given three segments whose lengths are: $\displaystyle 1,\;a,\text{ and }b,$
construct segments of lengths:

$\displaystyle (1)\;a + b \qquad (2)\;|a - b| \qquad (3)\;ab \qquad (4)\;\frac{a}{b} \qquad (5)\;\sqrt{ab}$

$\displaystyle (3)\;ab$

On a horizontal line, measure off: .$\displaystyle OP = 1,\;PQ = a$
Code:


o - - - o - - - o - - -
O   1   P   a   Q

Through $\displaystyle O$ draw a diagonal line.
On the diagonal, measure off $\displaystyle OR = b$
Draw $\displaystyle PR.$
Code:
                              *
*
*
R  *
o
b   * /
*   /
*     /
o - - - o - - - o - - -
O   1   P   a   Q

Through $\displaystyle Q$ construct a line parallel to $\displaystyle PR$, cutting the diagonal at $\displaystyle S.$
Code:
                              S
o
x   * /
*   /
R  *     /
o       /
b   * /       /
*   /       /
*     /       /
o - - - o - - - o - - -
O   1   P   a   Q

Then: .$\displaystyle x \,=\,RS \,=\,ab$

Proof
From similar triangles: .$\displaystyle \frac{b+x}{1+a} \:=\:\frac{b}{1}$

And we have: .$\displaystyle b + x \:=\:b + ab \quad\Rightarrow\quad x \:=\:ab$

3. Originally Posted by ReneePatt
Given three segments whose lengths are 1, a, and b, construct segments of length ... $\displaystyle \sqrt{ab}$. These are Euclidean constructions.
1. Draw a line with the length (a + b)

2. This line is the diameter of a circle.

3. Construct a perpendicular line at the end of a = begin of b. This line intersect the circle line. Construct a right triangle with (a + b) as hypotenuse .

4. According to Euclid's theorem you have in a right triangle:

$\displaystyle a \cdot b = h^2~\implies~\boxed{h = \sqrt{a \cdot b}}$

4. Hello, Renee!

$\displaystyle (4)\;\frac{a}{b}$

On a horizontal line, measure off: .$\displaystyle OP = b,\;PQ = 1$
Code:


o - - - o - - - o - - -
O   b   P   1   Q

Through $\displaystyle O$ draw a diagonal line.
On the diagonal, measure off $\displaystyle OR = a$
Draw $\displaystyle PR.$
Code:
                              *
*
*
R  *
o
a   * /
*   /
*     /
o - - - o - - - o - - -
O   b   P   1   Q

Through $\displaystyle Q$ construct a line parallel to $\displaystyle PR$, cutting the diagonal at $\displaystyle S.$
Code:
                              S
o
x   * /
*   /
R  *     /
o       /
a   * /       /
*   /       /
*     /       /
o - - - o - - - o - - -
O   b   P   1   Q

Then: .$\displaystyle x \,=\,RS \,=\,\frac{a}{b}$

Proof

From similar triangles: .$\displaystyle \frac{a+x}{b+1} \:=\:\frac{a}{b}$

And we have: .$\displaystyle ab+bx \:=\:ab +a \quad\Rightarrow\quad bx \:=\:a \quad\Rightarrow\quad x \:=\:\frac{a}{b}$