Question

How many arcs do we need to construct a ${45}^{\circ }$ angle using a ${90}^{\circ }$ angle if we are given only a line segment OB?

Answer 1

To construct a ${45}^{\circ }$, we need to follow the steps given below.

1. Draw a line OB of given length.

2. With O as the center, make an arc of any radius intersecting OB at X.

3. With X as the center, draw another arc keeping the radius the same intersecting the previous arc at D.

4. $\angle DOB$ will be ${60}^{\circ }$.

5. With D as the center and the same radius, draw another arc intersecting the first arc at C.

6. $\angle COB$ will be ${120}^{\circ }$.

7. Bisect COD drawing two equal arcs from each point intersecting at E.

8. Join EO and extend till A. $\angle AOB$ will be ${90}^{\circ }$.

9. With O as the center and any radius more than OX, draw another arc intersecting OA and OB at Q and P, respectively.

10. Bisect AOB drawing to equal arcs from P and Q points intersects at F.

11. Join OF. angle $\angle BOF=\angle AOF={45}^{\circ }$
Therefore, we need 8 arcs to construct a ${45}^{\circ }$ angle.