How can I draw a line with length "a^2"?

Question

I have a line whose length is "a". Can anyone explain procedure of drawing following lines?

1-line whose length is "a^2"

2-line whose length is "sqrt(a)"

Answer 1

As Rahul wrote, you need to have a line of length 1.

Once you have this, you can use similar triangles to solve $\dfrac{1}{a} =\dfrac{a}{x} $ so that $x = a^2$.

To find $\sqrt{a}$, draw a circle with diameter $1+a$ and erect a perpendicular to a diameter a distance $1$ from the end. Where this intersects the circle, it makes a right triangle with the diameter, and the altitude $x$ satisfies $\dfrac{1}{x} =\dfrac{x}{a} $ so that $x^2 = a$ or $x = \sqrt{a}$.

Nothing original here - it's all in Euclid.