Question

Question 84

Draw an angle of ${60}^{\circ }$ using ruler and compass and divide it into four equal parts. Measure each part.

Answer 1

Steps of construction are as follows:

Step I: Draw a line segment $\overline{PQ}$ and mark a point O on it.

Step II: Place the pointer of the compass at O (as center) and draw an arc of convenient radius, which cuts the line PQ at a point X.

Step III: With the pointer at X (as center) and same radius, draw an arc that passes through O, which intersect at Y.

Step IV: Join OY and produce it to B. We get $\angle BOX$ whose measure is ${60}^{\circ }$

Step V: With O as a centre and using compass draw an arc that cuts both rays of $\angle O$ at X and Y.

Step VI: With X as centre, draw (in the interior of $\angle O$ an arc, whose radius is more than half the length of XY.

Step VII: With the same radius and with Y as centre, draw another arc in the interior of $\angle O$. Let the two arcs intersect at D, cutting arc XY at L. Then $\overline{OD}$ divides the $\angle XOB$ or $\angle QOB$ into two equal parts.

Step VIII: Now, taking X and L as centre having radius more than half of length XL, draw two arcs respectively which cut each other at R.

Step IX: Join $\overline{OR}$, which divides $\angle XOD$ into two equal parts.

Step X: Now taking Y and L as centre, having radius more than half of length YL draw two arcs respectively, which cut each other at M.

Step XI: Join $\overline{OM}$, which divide $\angle BOD$ into two equal parts.

Thus, $\overline{OM},\overline{OR}$ and $\overline{OD}$ divide $\angle XOB$ or $\angle QOB$ into four equal parts.