Question

Find the square root of 4.2 geometrically

Answer 1

On a line, draw from left to right a segment 1 unit long, then extend it farther to the right with the next segement, but make it 4.2 units long. The two segments, united, make a longer segment: on top of this longer one, as a diameter, construct the semi-circle. From the point where your two original segments meet, call that point "P", construct a perpendicular segment up to the semi-circle, and it's length will be the square root you're seeking. That's because the three segments from P are in the following ratio: 
L for the segment to left of P, U for the one upwards from it, and R for the one to the right of it, 
L/U = U/R whence LR = U^2, but since L = 1, R = U^2, and so U is the square root of R. 
And all that, in turn, works because those three segments, and the diagonals connecting their ends, form a set of three right triangles that are similar: a large one on the diameter, and the other two its parts. 
And that in turn happens when you construct a triangle upon a diameter, reaching to the semi-circle, it's a right triangle; and when you divide a right triangle like that, the two parts are similar. 

Answer 2

DrawAC=4.2

Take midpoint as O

Now Draw a semicircle from A to C

Now,take a point O from b of distance=4.2-1/2=3.2/2=1.6

now make a prependicular BD

now BD is underoot 4.2