Compass and ruler construction Given the three line segments below, of lengths a, b and 1, respectively:
construct the following length using a compass and ruler: $$\frac{1}{\sqrt{b+\sqrt{a}}} \ \ \text{and} \ \ \ \sqrt[4]{a} $$
Make sure to draw the appropriate diagram(s) and describe your process in words. We are also to use the following axioms and state where they are used:


*

*Any two points can be connected by a line segment, 

*Any line segment can be extended to a line, 

*Any point and a line segment define a circle, 

*Points are born as intersection of lines, circles and lines and circles

 A: I don't want to answer for you but I'll give you hints.
Forget constructions for a moment.  How would we represent $\frac 1k$ and $\sqrt{k}$ geometrically?
As $k$ won't necessarily correspond to any integer or rational value, the only way I see off the top of my head to do ratios is via similar triangles.  
Can you construct two similar triangles $ABC$ and $XYZ$ ($A$ corresponds to $X$, $B$ to $Y$ and $C$ to $Y$)  so that $ABC$ has sides of length $BC =a$ and $XYZ$ has a side of length $XY = 1$ any $YZ = AB$?
That is $AB$ ~ $XY= 1$ and $BC = a$ ~$YZ=AB$ and $AC$ ~ $XZ$.  
If so that would mean $\frac {AB}{BC=a} = \frac {XY = 1}{YZ=AB}$ so $AB^2 = a$. So $AB = \sqrt{a}$.
So how can we construct such similar triangles?  Well, maybe $YZ$ should be $AB$.  Thus two triangles $ABC$ ~ $XBA$ where $BC = a$ and $XB = 1$
So construct a line $CBX$ so that $CB=a$ and $BX = 1$.  Can you construct a point $A$ so that $ABC$ ~ $XBA$?  What if the line $\overline{BA}\perp\overline {CB}$ and $\overline{CA}\perp \overline {AX}$?
Finding a line $\perp$ to $\overline {CX}$ and $B$ is straightforward because we have the points.  But finding an unknown point $A$ so that $\overline{CA}\perp \overline {AX}$ may not be obvious.  But Hint: given a semi-circle any angle from diameter endpoint, to point in the semi-circle, to the other endpoint will be a right angle.
So that $\sqrt{a}$.
To find $\frac 1a$ is easier.  Find similar triangles $ABC$ and $XBA$ as above where $BC = a$, $AB= 1$ and Then $\frac {AB=1}{BC=a} = \frac {XB}{BA = 1}$ so $XB =\frac 1a$.
To construct those triangle just create $ABC$ be making a $\perp$ to a line 1 at point $B$.  Mark off $C$ one line so $BC = a$ and $A$ on the other.  At $A$ construct a $\perp$ to $BC$.  Where this perpendicular line intersects the extended $\overline {BC}$, mark as $X$.
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