Given a segment of length a, show two different ways for constructing a segment of length (i) (square root of (19)) * a and (ii) a * (square root of (2/3))
Start by assembling a checklist of techniques for constructing segments whose length is some multiple of $\displaystyle a$. For example, can you construct one with length $\displaystyle ka$ (where $\displaystyle k$ is a whole number)? What about $\displaystyle a/k$? ... or $\displaystyle \sqrt2a$? What about $\displaystyle \sqrt ka$ for other values of $\displaystyle k$? (Think of a right-angled triangle in which two sides are known multiples of $\displaystyle a$. What will be the length of the hypotenuse?)