Question

Draw a line segment AB and by ruler and compasses, obtain a line segment of length $\frac{3}{4}$ (AB).

Answer 1

Draw a line segment AB using the ruler.
(i) With A as centre and radius more than half of AB, draw arcs on both sides of AB.
(ii) With the same radius and B as centre, draw arcs cutting the arcs drawn in step (i) at P and Q.
(iii) Join P and Q. PQ intersects AB at C.
(i) With A as centre and radius more than half of AB, draw arcs on both sides of AC.
(ii) With the same radius and C as centre, draw arcs cutting the arcs drawn in step (iv) at R and S.
(iii) Join R and S. RS intersects AB at D.

Bisect AC again and mark the point of bisection as D.

So, we have:

AD = $\frac{1}{4}$(AB), DC = $\frac{1}{4}$ (AB) and CB = $\frac{1}{2}$(AB)

∴ DB = $\frac{1}{4}$ (AB) + $\frac{1}{2}$(AB) = $\frac{3}{4}$(AB)
Thus, DB is the required line segment of length $\frac{3}{4}$(AB).