Question

Construct an angle of ${90}^{\circ }$ using ruler and compasses and bisect it.

Answer 1

Steps of construction:
1. Draw a line segment AB.
2. With A as the centre and a small radius, draw an arc cutting AB at M.
3. With M as the centre and the same radius as above, draw an arc cutting the previously drawn arc at N.
4. With N as the centre and the same radius as above, draw an arc cutting the previously drawn arc at P.
5. Again, with N as the centre and a radius more than half of PN, draw an arc.
6. With P as the centre and the same radius as above, draw an arc cutting the previously drawn arc at Q.
7. Join AQ cutting the arc at O and produced it to C.
8. With O as the centre and a radius more than half of OM, draw an arc.
9. With M as the centre and the same radius as above, draw another arc cutting the previously drawn arc at
point R.
10. Join AR.
Thus, AR bisects $\angle BAC$.